So the class we're taking is it's a multicultural counseling class there's about 12 of us in it at the moment and we're just trying to go through different cultures that we're not maybe familiar with in order to try and I guess more familiarize ourselves with it and maybe try and see from different perspectives any potential issues or maybe conflicts or biases that we may encounter in future clients and ourselves. So I was wondering, since you have mental health experience do you have any advice on that? Great well first of all thank you for you know for doing something like this because it shows your interest and you know and trying to understand and bridge the gap. Cultural competency is really important and it is something that's critical for mental health providers to know about because in many cases it can prevent care right if someone is has predisposed ideas or or biases as you said or just prejudices or or just lacks understanding they may not even feel comfortable to seek help and so if we can somehow address those concerns then we can hopefully encourage people to be more open to seeking out services. So I want to applaud you for doing something like this. As far as yeah some of the barriers for access of care that we've come across from our community because it is something certainly we've witnessed and we've had to grapple with in our community that immense stigma with mental health. When I started for example back in 2009 when we started our website Mental Health for Muslims I started with my cousin who's a clinical psychologist and then I had a lot of experience in the community because there was such a stigma to seek professional help they would seek out people like myself or other religious leaders in the community assuming that whatever they were going with or through whether it was their own personal struggles with anxiety depression things that we knew symptomatically had you know we already knew how to define those things but because of their lack of knowledge of these were mental health issues they assumed that they were spiritual crises. So that is for the Muslim community a big issue where a lot of people self-diagnose and presume falsely that what they're presenting is a lack of spirituality there's a disconnect there so they sometimes are so turned off by the idea of actually seeking professional help because in their worldview that's just not a possibility it has to be a spiritual explanation for why I'm sad and depressed maybe I should pray more maybe my maybe God is not happy with me so they'll make all these deductions which are of course and most of the time not there's no basis to them other than their own lack of awareness so we found that that's a big challenge where a lot of times the stigma to mental health is that people associate that you have you know it's it's unfortunate but there are a lot of people who equate mental health issues with some intelligent deficiencies or other character quality deficiencies not realizing that it's just the same as if you you know had a health problem you wouldn't presume anything about a person's character or their you know their their who they are if they had a health issue but unfortunately when it comes to mental health that's the stigma right like oh that that person is not well they're crazy God forbid you know they use these sort of terms that are really just not right at all but that's the that's the conclusion that's often made that mental health means that someone's not right and and they're almost irreparable so I don't want to associate myself with that at all it has to be a spiritual issue that I can find a spiritual solution for that's probably the biggest barrier that we've come across within our own community but we have made some strides because since we started that website in 2009 with our very few practitioners and these conversations even in the community weren't really open we have come like almost light years in the past you know how many years now has it been right 11 or so years I have seen in my own personal experience so many more conferences events programs at the mosque every Friday you'll see family nights and programs where imams even and people who are typically you know who in the past may not have been too open about this topic now it's a very common thing where mental health issues are brought up as community wide you know program that the programming that that is important for everybody to hear so they'll put it you know sometimes even in the Friday sermon they'll have addresses that talk about domestic violence substance abuse depression anxiety all of the very common disorders that we're seeing on the rise as a people as a species right they're they're addressing those to the community so we have made some some really great strides we've all we've also seen a huge surplus of more practitioners there's more there's something I don't know if you're familiar with but you may have heard of it there's the Khalil Center which is probably the leading national now organization of counselors that have you know offices all over the US and a branching out I think even into Canada where they are really taking the lead in the in the discussions around mental health there's there it's basically a service center but they have chapters throughout the US and they provide services mental health services to the community so there are organizations like that and many others that have come up within just I would say the past 10 to 12 years it's been really incredible to witness yeah that's great sounds like there's it's really growing the support so being hopefully new therapists or counselors what could we do to help try and also bridge that gap so they feel accepted or they're able to receive help that's a great question again I thank you for asking that question I think there should be some sort of cultural competency training about Islam and Muslims first before someone even considers taking on Muslim clients because the issues that do come up oftentimes are so in meshed with their spiritual identity as or their religious identity as Muslims so it's very hard to divorce those things in in a therapy session if and that's also one of the barriers that we found is that people are reluctant to go to see non-Muslims because they think they're not going to understand what I'm going through or why this particular incident is affecting me the way they are as opposed to a Muslim who at least has that context they understand right so I think when you have just a very simple like you know primary you know refresher about Islam to know the beliefs what the beliefs are what the the daily you know life of a Muslim entails will give I think the provider a really better lens to understand why certain things may affect their client that that maybe is not understood by by or seen in other people of different faiths and backgrounds because for Muslims Islam is not just a religious identity that we ascribe to on certain days or around holidays or you know we just kind of it's a label or that something that we it's actually a lifestyle so Islam very much affects everything that we do all the choices that we make our world view so if you don't have that understanding then it might be difficult for you to understand why someone you know won't take your guidance on well have you tried this or have you tried that because in their particular worldview that's just not an option like for example divorce you know I as someone who's been doing spiritual counseling for a long time for Muslims keeping the marriage intact is so important and it's we take marriage very very seriously divorce is permissible in the event that the marriage is just not salvageable but you really do try your best to save the relationship especially when children are in the mix so that is such an essential you know belief that that is very rooted in many Muslims and their just values so if you have a practitioner who doesn't have that same understanding and they will in the very first you know session suggest that maybe you should just dissolve the relationship you know if there's a it's like a counseling session for example you guys just don't seem very compatible you fight a lot there's too many other variables you know finances in laws and so they just deduce quickly that maybe the marriage is just run out of you know like there's really no point because in the Western or the you know their context it's just seems like that's the most practical thing to do then the client will likely feel like okay see they're just quick to rush us to divorce they don't really care about trying to really help us and that is actually a very common complaint I've heard from Muslim clients who have gone to see non-Muslims they always come back saying no they're too liberal minded or they don't have the same family values same understanding of like even in laws you know a big issue that you'll find if you ever are to have clients in the Muslim community are the families are very enmeshed in Muslim marriages so it's not so simple to tell or it's not easy to tell your client well you should just create boundaries with your mother-in-law and tell her not to come over you know those kind of things and in our Western sensibilities they make sense because it's like boundaries are very clear for us and we're more empowered to have that sort of those lines you know really clear but for Muslims it's very difficult because it will absolutely impact the entire marriage you know if you were to draw such clear lines with your mother-in-law or your father-in-law it could potentially lead to a divorce it's that serious right so I think having that knowledge that there's certain values that you know people of Muslim you know background or what regardless of the culture it's pretty shared because of their faith that they adhere to strictly therefore I have to kind of adjust my tone or my approach to not just you know dismiss those things or to overlook those things in my in my in what I provide and the service that I provide so yeah you've been mentioning a lot about family and I know how important it is but there's also it seems like almost a community like family within the faith itself like it's it may be separate from like the you know the family family but it still feels like they that it's treated as kind of a larger family within the faith absolutely that's a very very on point we do the community is very we do see ourselves as a one large body although we're definitely not a monolith there's a lot of variety but we are it's part of our belief system to see ourselves as one body so we call for example the Muslim community at large is called the umma which is like a nation of Muslims and so there's a lot of you know teachings in the faith that say that we do we should really try to strengthen our brotherhood our sisterhood so if you're part of a larger community wherever you live yes you see your community members as extended family members in many cases in there there is of course we have you know there's there's lines I mean it's not that they would know all of your private you know information we're not like that but it's to you do take into consideration the larger impact of your choices and decisions as an individual on the on the community as well we're not so separate that we just don't care like for example reputation is a very big deal you know in many of our Muslim cultures like so a lot of people will take that into consideration like how is that how is the community going to view me you know if I take this decision or if I take this decision so sometimes those factors are also really important and why they may choose to go one way or another and if a provider isn't aware of that and again they give very Western ideas to people who are not defined by those ideas it may cause just the lack of you know connection and then you might not see that that client again because they just feel like well they don't get me they don't get me they're just giving me very Western perspectives about things and I just I don't agree with that that's not my philosophy or that's not my viewpoint so but it can be you know bridged by just having more understanding and education about those things because I've actually sent friends of mine who didn't have success or couldn't find Muslim therapists in their area we've done some research and looked for people who in their you know description of their services they actually do describe that they do have competency around certain cultures and they found great success like that person did their due diligence or their you know they did their part to learn more about different cultures and it did help the way that they were able to provide service I think it's it's so it is possible to do that yeah so in your experience do when you're dealing with clients do you incorporate the family a lot more or is it still kind of an isolated thing that you just kind of use it use your knowledge and experience as a background that's a good question I always do like to know I mean depends on the circumstance I've worked with individuals I've worked with you know actual families where the mother the father and the children are all kind of there and then couples so I think it just depends on each situation but if I feel that there is some involvement you know metals some behavior for example in a in a marital context then yes I will want to know the you know more specific details about the family and the relationship that the individuals have because you know there are some filial piety for example in Islam is very important it's one of the top you know beliefs or tenets of the faith is that we worship God and we you know we have our own you know there's a lot of ritualistic practice that we have but one of the most important things that a Muslim must do or should do is to show reverence and respect to their parents so we that's a very big part of the Muslim you know mindset so it's a you know filial piety just having that so that can sometimes though posed you know issues when it comes to the marital relationship because a spouse may be completely conflicted if they have a very over a varying parent who is dictating to them how they should be in their marriage and you know everything that they should be doing in their marriage because they don't have good boundaries then the child or the spouse may feel really torn because they feel that they're doing something spiritually wrong like if they go against their parents right so this is where someone like myself who has a spiritual background can actually give them a balanced view that you don't have to pick your wife or your husband over your mom or dad right it doesn't have to be like that you can actually have compromise and maybe it's your mom or your dad that needs we need to bring them in and talk to them about hey listen you know we get that maybe you're having a really difficult time you know with your own personal situation or because you know a lot of our the older generation they get they create some unhealthy codependencies sometimes because being here in a foreign land not having their family from back home they can create those codependencies with their children that be that get very unhealthy and they're not aware of it and so sometimes having a third person like myself or maybe a religious leader who can come in and tell the mom and dad like listen yes you have rights in Islam and your children should be dutiful and respectful to you but you also have to respect their boundaries and this is what that entails so we can help to kind of you know dissipate the situation that way but there have certainly been instances where yes you have to involve the family if the relationship or the individual them because there's been individual cases as well where the parents are just not understanding you know the conflict that their child has individually and they need more education on mental health issues for example like depression anxiety what does that mean how can you better help them to you know help themselves instead of you know just you know shaming or blaming or kind of getting into that negative space because you're frustrated with your child or you're frustrated with this individual so sometimes it is necessary to get the family involved. We also mentioned using the imam with the parents as well is that something you would think would be you know acceptable in the mental health field to try and incorporate the spiritual leaders? I think it's absolutely something to keep as an option in the event that you're dealing with people who are very religiously committed people because you know they sometimes have that tunnel vision or just you know they can't really see beyond what what you're telling them so to have an ally who's in the community a religious leader who you can outsource when needed or to bring into a mediation for example or some conversation to help to you know again dispel certain false ideas because there are there's there's a lot there's sometimes just cultural things that get confused or passed on as religious ideas that an imam or or even a female you know teacher could certainly elucidate and clarify that listen that's a cultural idea that's not religious right so if you are concerned about your spiritual well-being or doing the right thing let me just tell you that that practice is nothing to do with Islam and it's actually just you know from your culture so sometimes it can be very useful to a practitioner to have a person in that capacity to you know to help to outsource when needed like I said but I would I would definitely look to your community and see if there is an opportunity to create a relationship with the local mosque where there they might have existing counseling services and you could even work with and pair up with providers from the mosque or from the community center to to help with clients when you get them or to just you know consult on certain issues that's absolutely something you should I would recommend doing okay well and it's kind of in the same vein where should where could me and my classmates go to find more information so we can kind of further our understanding I know you have the website the mental health for Muslims calm but what other resources do you think would be viable for us that's great there's a lot of I mean you know if we're talking specifically about learning about Islam there's a great many many different websites or organizations one that comes to mind that I personally know and have worked with and I think they do a phenomenal job of explaining a lot of the core beliefs of Muslims and Islam in a way that Westerners totally get and understand because they that's what they do is an organization called the Islamic Networks Group and they're actually based here in California in the South Bay I worked with them for a long time but they have a national reach they're in pretty much all the major cities they do what they do is they provide trainings actually cultural competency trainings to all different institutions hospitals police station schools universities middle schools like really everywhere corporations that have called on them they will come and they will actually do presentations on or trainings that will give you know people more just general basic information about Islam really bring them up to speed about a lot of the things that we talked about and they also have great content on their website that you could like there's an FAQ most commonly asked questions about Islam a lot of things that people who have no idea about what Muslims believe in will will get and they I mean they work the office itself there's non-Muslims that work there so it's a very inner faith very just American organization that is trying to make Islam more reachable or you know more approachable for for people who are outside the faith so I think I can provide that in the chat if you'd like other website I can so you can look it up but I think they have great resources on their website yeah that'd be great yeah well another thing just from calling down and writing to the local Levenworth Center it there seems so open they mean they're welcome within you know a few hours I got an email back and then it you know phone call so it was amazing to see how open it is that we can just if we have questions or we just want to clarify or anything really absolutely and that's very much part of the Muslim tradition Muslims generally don't we don't do like missionary work for example we don't go out and we don't you know process size that way but we do invite and we're very open when people ask questions like you could go to any mosque and you know in terms of if your students or your fellow students were interested in observing I'm sure they would arrange a meeting where you could I mean right now with COVID it might be a little different but post you know when things open up or even online they might have programs that are for the general community or for anybody to tune into I'm sure they would be more than happy to give you access to that and generally that's that's the experience of a lot of people have they're usually surprised like oh wow I could just go in and talk to people and sit and watch the prayers and nobody says anything I'm like yeah because we're we have nothing to hide and we're very proud of our faith and you know people are interested we're very honored by that so we're happy to accommodate and even like with Manir and myself we were very happy to you know talk to you because it's it's actually really nice to have people who are interested in learning from sources instead of just turning on news media or going to random play you know places that speak for us whereas we can speak for ourselves in these circumstances so we're really happy to do that so thank you well thank you I know you got to get going here in a couple of minutes but I just want to say thank you and do you have anything you would like to say to our my classmates well first of all again I want to thank you Eric for coordinating this and all of you for being so sensitive I think really just having open conversations and not being I know that and I'm a very American I mean I don't know if you can pick up I hope I mean I didn't explain but my back although I was born in Afghanistan I was really raised here I have no memories of again so I very much consider myself American and I think I know that part of American culture is political correctness and we're sometimes so worried about offending people and like oh I don't want to ask too many questions or you know it's religious especially around beliefs right we don't want to politics beliefs are certain things we don't talk about openly in our society but for Muslims we actually really do welcome those questions and I think you will find that experience as we just said across the board with Muslims we're actually happy to answer questions so feel like free or to do that and I will make myself available for anybody who has follow-up questions to this interview I can send you my email address I do a lot of interfaith work I speak publicly nationally so I'm very comfortable talking to people I have no problem answering questions even if you think like that's a dumb question or should I even ask don't worry you know people about my hijab about women and women's rights I do entire presentations on women's rights in Islam I talk about interface of sight there's really no topic that I would feel in a little intimidated by the answer so I think if you if they're interested I can certainly make myself provide my contact information for specific questions and also for resources because our time is limited here I don't have that off the top of my head but I can make perhaps like a resource sheet with more resources for you and and even look to registries that I'm familiar with to see if there's something more to your locality that that might help you in this regard so I can do that as you know after this offline yeah that'd be amazing do you mind if I share your I have your contact info your email notes can I share you know what if I can give you actually this email address the other one is for personal use but the one that I'm posting right there that's more for general and I just it helps me to filter you know between family and business and all the other stuff that I get so yeah this second email address is is please feel free to share it with your students and and anything I have I'm on line also I'm on Instagram on Facebook I'm on Twitter so people are on social media my tag is the same and I do write a lot so I have a lot of talks that I give but I also write a lot so you can check out posts if you're interested kind of seeing you and I write on mental health advocacy so that's perfect I really appreciate this again but I guess we'll wrap it up now since you got to get going here in one minute yeah yeah in a couple of minutes unless you had any final questions I can I can stay on for a couple minutes I don't really have any leftover but so one quick one then because all of my classmates are female and you're talking about females in in Muslim community I think a lot of times they're viewed as weak or maybe not as equal but that's not true and I was wondering if you can kind of sure well I hope I'm example of that that we're definitely not shy timid and you know that's I mean those are just tropes and unfortunately the media has definitely stereotyped us down to just that but if you actually study Islam or talk to Muslim women you will see that that's couldn't be farther from the truth we're very empowered because you know looking at the history of Islam and Muslims 1400 plus years ago you'll see that a lot of the rights that Western women and women and other civilizations and societies have gone have received just in the past maybe hundred years or so Muslim women were given those rights you know centuries ago the right to vote the right to inherit the right to work the right to own property a lot of those things were given to us it was just part of our faith so we've had that for centuries so I think what we see though are is is cultural so a lot of I mean I come from a cultural like I mean culturally Afghanistan right where our introduction to Afghanistan happened after 9-11 it was put on the map and the instant images you see are people wearing that you know they call the the zoo key the beekeepers you know blue the the the Chador right which is you know an outfit that's traditional to the Afghan culture and it actually predates even Islam so people don't know that that that was just something that even after you know the liberation and the Taliban were dismantled and all you know the America and the troops and the Western troops entered you will still go to many places in Afghanistan and the women still wear that it's a cultural part of their identity so it's not part of Islam right but things like that that I think are really important for people to understand the difference between culture and the religion because the religion itself is very liberating to women you know we and I could give you I mean the presentations I do that's exactly what I do I go over the history I talk about historical figures I talk about all of the rights that are in that are in the scripture from both the Quran and the hadith which are the sayings of the Prophet Muhammad where we're showing clear example and evidence of the fact that women have always had rights in Islam and it's just a matter of systems right patriarchy is real and it definitely is you know in every corner of the world and when you see those systems in place then sure it's going to present as though you know the women may not have certain rights but not that's not from a religious standpoint and that's why you see Western Muslim women very much more vocal because they're not you know under a lot of those systems right we can be more vocal here so we find you find Muslim women and we have you know Rashida Tlaib we have Ilhan Omar they're in politics you know our first two Muslim women in political positions so if we were really like the stereotypes in a lot of the Hollywood films and all of these other you know areas where we're just we don't speak and we can't do anything we can't move without our husband's permission you would not see so many up-and-coming women who are very strong very vocal very outspoken who drive you know do things that again stereotypes may may say otherwise but we're proof that that's just not simply true and those are unfortunately just you know oppressive systems and government systems that are not in line with the faith but the presentation I have is actually I think it would be great I don't know if there would be an opportunity but if it's something you want to talk about offline maybe we can share some of those I know it's more you know with the whole class


So one thing that's not on there, which I announced whenever it was I announced it was Monday. So I announced Monday was that in general, homeworks will be due in class on Wednesdays and you'll get them back on Mondays in recitation. You don't have recitation next Monday because it's Labor Day. So this time you're on your own for the homework. My office hours I think are on there. There are Tuesdays from 1.30 to 3.30. And then also Friday mornings from 11 to 12 over in the math on the P level of the math building where my primary function is to advise students who want to be majors or mess with their classes or whatever. But I'm there so you can talk to me if you have a question. We don't do that anymore. So the calendar has to be readjusted so that there is exactly one Monday holiday, one Tuesday holiday, one Thursday, one Wednesday, one Thursday, one Friday so we don't have to shuffle days. So next Tuesday is Labor Tuesday because we need a Tuesday holiday so it's the one after the Monday holiday. So there won't be any. Well today your calendar says it's Tuesday but it's really Friday. None of that is here. Other questions? Nope? Okay. So as I said before I'm trying, I'm going to take this. I will hope that this will function properly. I'll put it up on the website. It will probably take at least a day and probably a couple of days for me to edit it and do the jumps I need to do to make it web-able. So if you want to hope that it will work, that means that you don't need to take as careful notes as you would otherwise or you could do what I do and that should never take any notes and then read the book and listen and think. I strongly encourage you to pay more attention to what's going on and work on understanding as it's going rather than trying to get down everything I say because most of what I say is in the book in a different form and I hope there will also be the video of it. So it's much more important that you focus on following what's going on rather than taking careful notes. For some people taking careful notes is how you follow what's going on so that's fine. Whatever. I'm not shy. Don't come over to my house and take a picture while I'm in the shower. Any other questions? There is a homework assignment. It's not really long. I don't think it's really hard. That is due one week from today and yes, that's like seven days, whatever, seven times 24, that many hours. You could do the arithmetic better than me. Okay. So I'm going to pick up, remember what I'm doing?  So if you remember what we're trying to do at this point is we're trying to recast some of the standard notions that you were familiar with in algebra slash Cartesian geometry, whatever so that it works in more dimensions. And what we're doing here to generalize, say the notion of a line which in two dimensions we can write as y equals mx plus b if you want. I don't actually like that formula so let me not do it that way. Let's write it as y minus y naught equals m times x minus x naught because that has a little more meaning. So this is a review. So anyway, so that means that we have some point x naught y naught that sits on the line and we can recast this in a slightly more general form where we think of this as really saying, this is really saying go to the point x naught y naught and move off in the direction that goes up m units where every one unit can go over and that's just a direction and just goes some distance in that way. So this is just saying the same thing in a different way. We go to this point and then we move in that direction some amount and that gives us everybody on the line. So this red thing can be viewed as the vector which I could write this way, let's see that's the y I could write it as a column vector like this I won't do that mostly because it's tedious which I might also write it this way which says that when I go one unit in the x direction I go m units in the y direction so either way and really let's just think of this as some vector v and so now I can rewrite this line as saying start at the origin go to the point x naught y naught and then move some distance in the direction of the vector v so every point on this line is really can be viewed as points of the form x naught y naught plus some number t which is how far I go in the v direction this is the same thing just another way to view how this line can be described so if I want to think about this point over here it's go there and then move it looks like about four times the length of that vector here and if I want to think about this point over here on that line it's go there and back up about two and a half times the length of the vector so that describes all of these points these are of course vectors but we think of them as vectors that point from the origin to that point on the line and the advantage of doing that instead of using this formula is that that notion works just as well in any number of dimensions so that if I'm working in three dimensional space and I've got some point here sitting out in three dimensional space given three coordinates x naught y naught that's supposed to be z z naught I can play the same game I say go to that point as this vector and then from that point pick some direction that you want to go about this way and then just take everything that moves in the same direction as this vector by some scale of that so this line which is a little harder to visualize in three dimensions I guess it punctures the plane just about here so that's below in my visualization of what's going on here this describes that line because it just says go here and then move that way by some amount of t so if we call this vector v again then these guys are exactly the same form this line is the stuff of the form in fact let's call this guy I don't know a this is everything of the form a plus t times v and this will work in any number of dimensions if I'm in a 17 dimensional space which is a little hard to draw I will have a vector in r 17 plus t it's a direction in r 17 and that will describe everything on that line let's go there and then move off that way and this is sort of naturally how you might think of a line as well go down to the beach and aim your camera that way and what do you see? everything there it's exactly the same okay so this is what we did last time let's just see if anybody understands so what's the equation of the line through the point let's take two points 1, 2, 3 and 4, 5, 6 so if you understand you should be able to answer that question or at least walk me through how we do that so don't just tell me the answer tell me what you do yeah well first you want to find that that is between the two points okay so let me draw a cartoon of what's going on here's 1, 2, 3 here's 4, 5, 6 and I want to find the vector between them okay so what's the vector between them? get that one okay so that should be like 3, 3, 3 wow okay that's just amazing how that worked out so this vector is the vector coordinates 3, 3, 3 now you can use either the two points is your initial point okay plus t times the vector 3, 3, 3 okay so this will be let's use 1, 2, 3 1, 2, 3 plus t times 3, 3, 3 good anybody confused about that? now we can write this in a slightly different way we can put them together this is 1 plus 3t 2 plus 3t 3 plus 3t so if we don't like vectors you can see that we are actually describing this line by telling you all x coordinates of the form 1 plus 3t y of the form 2 plus 3t z of the form 3 plus 3t so we can also describe this instead of in vector form as x is 1 plus 3t y is 1 2 plus 3t and z is 3 plus 3t so those three equations together also are another way to describe this line it's the collection of points that look like that these are really exactly the same just written out in a slightly different form so you pick the t value and then we know where we are in the line there's actually a little more information here than just the line there's sort of an implied speed that if I increase t by 1 unit then these guys increase by 3 units and so this is the same line in terms of the set of points this line let me write it in vector form 1 plus s let's use t they're not the same t 3 plus t 3 plus t they're not the same t so I really should use I don't know, t1 I don't know why I'm using t but I am this is the same line in terms of points but the speed is 1 third slower if I increase t by 1 unit I only move a third of the way to 4, 5, 6 whereas here if I increase t by 1 unit I move all of the way to 4, 5, 6 but since we're only paying attention at this point to the collection of points that are described by this these are the same line just like you can drive down the LIE at 30 miles an hour you'll probably get her ended, but you can or you can drive down the LIE at 80 miles an hour you're traveling the same path but you'll get there a lot quicker at 80 miles an hour unless you get arrested so you can do either one and they're the same path but at different speeds and so sometimes you care about the parameterization the way you're describing this collection of points and sometimes you just care about the collection of points so here the speed is different and of course I don't need the same base point either I can use 4, 5, 6 as what's your name? Nick as Nick said I will maybe remember next time I see you that your name's Nick but at least forgive me if I don't but maybe by the end of the semester I'll remember unless you leave okay so any questions about lines stuff with lines no okay so in two dimensions that's about all we can deal with but in three dimensions we can also think about planes so let's think about how to describe a plane so these are I'm only dealing right now with flat objects I'm dealing with the most simple geometric objects like lines and planes and hyperplanes and so on but so let's think about how you might describe a plane how many are is anyone not familiar with this notation you're not familiar okay so I will use this notation because well I used to it so the line so this is on the side the real line we use r for the real numbers because r go together and r2 is a copy of the real numbers in each direction and we consider all pairs of those so r2 is the plane which is pairs of reals and r3 is obviously space r12 is 12 numbers the set of all x1 x2 up to x12 such that x is important so it's all 12 couples of numbers so this is useful notation if we want to specialize to a specific dimension we also don't have to think about just reals in this way we can think about integers in this way so I don't know that I'm going to use it but let's just say it anyway z is the integers which is all the negatives 0, all the positive numbers someone and say z2 will be pairs of them and we can generalize this in many ways so also we'll do a little complex notation z is the complexes so this notation is fairly standard in mathematics but you cannot see it until somewhere around the university or college level high school you pretty much have to deal with real numbers or you deal with integers what the difference is good, it's a fraction thing let's do that q is the rational set of fractions q isn't 0 what else I guess they have no common divisor about the gcd z is for zealot and german and q is what is q at the bottom I don't know it doesn't stand for comptabular so those are the common ones that we'll use and typically we'll focus on rn n is not a half n is a positive full number although r0 looks like it's 0 real numbers not much to do with it but we probably won't deal much with r0 that's the wrong set let's go back to this what about a plane in three dimensions how can we describe let's draw a picture I like to draw pictures you can't read my pictures but I can so here's a plane it sticks in the back it doesn't look much like a plane anyway I'll try it I know what I'm drawing I don't care if you don't know actually so here's a plane and it's in r3 so there's a plane sitting in r3 somehow generalizing this notion how might we describe a typical point in the plane here right so I need a point in the plane that isn't this one I need to know how to get to the plane in the first place and then I can get to this point by just taking this vector but that will give me a line sitting in that plane so I need another vector to get everything else and then I can express that as a linear combination of these two vectors just like if we want to describe point on the map we have two coordinates tell us how far north or south and how far east or west here we can say how much in the green direction and how much in the red direction so I can get to this point here by saying go to the black point move in the green direction a certain amount and then move off in the red direction it's the same as that red direction it's parallel so direction doesn't have a base point it just has a way to go and these are supposed to look like the same one and maybe this is half of a red but whatever so one thing though we have to keep in mind is I could be stupid and choose my red direction to be parallel to my green direction and then I'm not going to get much of a plane so to describe the points in my plane I need some kind of a base point so I have my base point let's call it P I don't know why I switched from A to P but I did and then I have two dimensions to add on I have one direction that I go in the T and then I have some amount that's my, in this picture is my green direction and in this other way I have get somewhere on the plane move in the green direction some amount move in the red direction some amount and I need V and S no W to be not parallel if V is parallel to W I'm not going to get a plane I'm going to get a line because then I can collapse these two together and describe in terms of T plus S divided by the dimension but I can just describe this as a line this describes always a two-dimensional plane in any space that is three or more dimensions I mean it kind of describes a plane in the plane but there's only one so there's not a lot to do we can describe something in R2 by this 2 but every point in R2 is a point in R2 so we always get the same plane I have no idea if you understood what I meant by that but what I'm saying is if we're talking about it in R2 we can describe points in R2 this way as well I see it as a second thing take a base point go some red direction some amount and some green direction some amount and for as long as the red and the green guys live in the same plane any point on the board this way but every point in the board is always represented this way there's only one board to put it in okay I answer that did I answer it? no share do you have enough gum for everybody? I was just thinking about whether or not you could have a two-dimensional plane in four space yes you can so this will work just fine two vectors with four components and this will describe a two-dimensional plane in four space but the complement of that is also a big space right just doesn't divide four space in half in the same way that a line doesn't divide three space so if you wanted to do space three space exactly you would have three non-parallel vectors and if you wanted to describe some eight-dimensional space we have eight non-parallel vectors so depending on how much you have you can still have three non-parallel vectors yeah so when I meant non-parallel I meant non-planar non-planar non-parallel is wrong non-planar so I need them to be so the correct word which since we're doing the linear algebra here is that another way to say this is that V and W are linearly linearly independent and then the generalization works so if I have three vectors I need U, V, and W to be all linearly independent I didn't define what linearly independent means but you can't do the others yeah I can't express one in terms of the others so another way to say this so let's go off we'll get there by the end of the year so this is not yet what we're really doing but what the heck I'll tell you now so some collection of vectors V1, V2 up to the end so I have n vectors and these guys are linearly independent if we have let me write it this way A1, V1 plus A2, V2 plus up to An, Vn equals zero there's an equation if this is true means that all of the a's are all zero so the only way that such an equation can be solved is if all of the coefficients are zero then these guys are linearly independent so this is usually the standard definition for linearly independent but a something that you need to think about to see that they're the same and when we discuss this I'll go through it again this is the same thing as I can't write An, V in terms of the a's so this is a more mathy way to say it if you think about this for a few minutes or a few seconds or a few hours depending on how long it takes you to puzzle things out you'll see that these are the same and I can explain it to you in its few seconds but let me not let me leave it for you to think about this and you decide whether it's going to take you a few seconds or a few minutes or a few hours wait a few weeks until we cover the puzzle okay so that was my side remark so in general can't read what I wrote that means that is it guessed yeah do you have an arrow over the zero this is zero vector these are vectors you have arrows over them and this is a vector I'm going to miss these little vectors a lot so I should write them in boldface I'll take all the next A, V and V I don't think so okay so this is the definition of linearly independent don't worry about it if you didn't follow it it really means this so coming back to plane so if we have a hyperplane I don't know a K dimensional hyperplane is going to be and obviously it only works in a bigger than K dimension otherwise it's stupid a K dimensional hyperplane is going to be something of the form some vector P plus T1, V1 T2, V2 blah blah blah K, VK so I forget who asked about 4 plane or something so here we can describe a 4 dimensional plane in this way and here K is 4 so we stop at the end yeah up to K minus 1 no I have K because I started at 1 instead of 0 if you want me to start at 0 I'll go to K minus 1 my math is not computer science so I start counting at 1 he asked if it should end at K minus 1 but I need K vectors so if I start counting with 1 I stop at K to get K things if I'm a computer person I always start counting at 0 and then I stop at K minus 1 and if I'm really weird I might start counting at 7 and go to 7 plus K I mean that's how many numbers so when you say plane it's not necessarily a flat plane it could be like our space no it's a flat plane nobody call it a hyper plane because it's more than 2 dimensional so generally in English one calls something a plane it means a 2 dimensional space and so a hyper plane that's a K dimensional space because it's in a hyper space yeah it's in a hyper space the plane itself is more than 2 dimensions so it's not a plane in the sense of this floor I can think of this room as a 3 dimensional plane sitting in 4 dimensional space or a piece of a 3 dimensional plane where I'm in 4 dimensional space where the 4th dimension is time and if I just take a snapshot of this room at 1 instant in time that's really a hyper plane a 3 dimensional flat space sitting in 4 dimensional space so once it's it is a 3 dimensional plane you call it a hyper plane or a 3 plane yes hyper plane usually means a plane other than a 2 dimensional plane in 3 dimensional space so if you don't use the word hyper that's fine plane always implies flat so if it's curved it's not a plane is it curved? are we good? okay yeah this is in so this is in r m and here m is greater than or equal to k and each of the vectors v1 up to vk are all m vectors let me write it in column notation 1, 2, 3, 4 plus t times 7, 1, 6, 5 plus s times 8, 0, 7, 9 plus uh I need another letter r times 1, 1, 1, 1 this is a 3 plane in 4 space and if I add another number here 5, 5, 5 5 this is a 3 plane in 5 space but this 3 plane does not necessarily is not necessarily to be a flat plane what's flat mean? I understand that this is flat in an easy to make sense of way but in the same way this room is flat one can define and we're getting way ahead of ourselves the notion of curvature in more dimensions than 2 more than a curved thing we can define curvature we can talk about a curved space in a well defined way this is not curved at least not in a metric that we've written things it's not flat in the sense that you can ride your roller skates on it because it's a 3 dimensional thing it's a 3 dimensional thing sitting in a 5 dimensional world right so you can ride your roller skates and fly your airplane that's fine but you can't use your time machine yeah so if you're to square any one of those terms like t squared we'll have a different discussion it may or may not become curved hands on and stuff so we'll talk about that more but for now we're just thinking about the easy stuff the flat stuff just like before we started doing calculus here you think a lot about lines for I don't know from junior high school to early high school you think about lines and only after a while you say hey it could be prevalent can you talk about circles okay so we have that and I'm like when do we stop 5.20 so I'm not waiting I'm just a bill alright so we have these geometric objects and here lines and planes are kind of the same kind of thing just a matter of extra dimensions and things like that so for the next class or two we're going to mostly focus on lines and planes and then we'll move to more complicated objects once we've mastered that so but actually let's step back a little bit let's talk some more about vectors so if you remember I can add vectors and when I do this I just add the subtractive components or geometrically if I have a vector v1 and a vector w if I want to add them together I just take the v I stick it on the n's w and then this new vector v plus w which just corresponds to adding together the coordinates which is just saying go in the w direction until you get to the n now travel in the direction of v you get to its n and in the n you've gone to v plus w so this is fine I didn't mention but of course somebody already mentioned for me we have a special vector called the zero vector which is just all zeros and so we have some algebra of vectors we can add them subtract them vectors have inverses we have a zero vector yeah because it looks like this so the zero vector is don't going on and here I'm being a little sloppy I'm implying that we're at a fixed number of dimensions there's only one zero vector for each dimension just like there's only one vector one one one can you also say that any vector has an ending for a zero from the end of it so it's the same thing you can invent the vectors in a higher dimension by sticking zeros on the end but what I mean by the zero vector so let's be a little more algebraic about this so we could I could write all of this stuff in the way that if you were taking math 310 or maybe even math 2.11 where we could write some properties which describe vectors in the way they operate together one on the other we're saying that we have some vector V so I can give an axiom that says that for any vector V there there's another vector let's call it negative V so that V plus negative V and there's a unique vector zero so I'm just saying that I have lots of vectors one of them is special I call him zero and anytime I have a vector he has a friend who I call negative V so that when I add them together I go to zero this is similar to say in the plane there's an origin there's a special point in the plane which is where the axons cross and if we think of this as it's a collection of all vectors that's the zero vector now it doesn't necessarily have a unique representation in that I can write it as 111 minus 111 still a zero vector but I have a special die called zero so it would be proper to say that 11 plus negative 181 big zero zero with an arrow or bold zero so often I will be sloppy and everybody will be sloppy and Nick may be cockney on this I just wrote zero and I didn't put an arrow this is false this doesn't even mean anything it needs to be a vector for it to mean something because if I add up a bunch of vectors I better get a vector back I can't add two vectors and get a dog I add two vectors and get another vector so there's a lot more properties that describe vectors like given two vectors if I add them together this is another vector so just like numbers if I take two numbers and add them together I get another number this is true of real numbers but it's not well it's not true of numbers less than 10 if I add together two numbers less than 10 I don't get another number less than 10 necessarily I might sometimes but it's not closed under addition but vectors addition of vectors is closed and I can describe it in terms of the little arrows I can describe it in terms of writing in coordinates and add the coordinates together they're the same thing so if I add two vectors and it's also it's the same as W plus V that is vector addition is community now I don't need you to memorize all these properties but I'm just writing them down vectors work the way you would expect but then in addition to having that as I mentioned before we also have another collection of numbers laying around so if you take 308 we'll definitely get back to this in a more formal way I mean kind of informal about how vectors work now but we have in addition to our collection of vectors we have scalars so the scalars they need to be in a field and in our case our field is real and that is given a vector V some vector V in my vector space and given some scalar space and given some scalar space and given some scalar space and given some scalar space and given some scalar R in the reals then our field is a well-defined vector G and it works the right way we can prove these facts but I'm not going to and similarly there's a special scalar so the number one times V is joint V so we need that if you multiply by one nothing changes otherwise that's kind of how one should work and zero times V should be the zero vector now these properties are not all will be independent but don't worry about it what else do I need and I have a distributive law right if I take R and I take S as scalars and I scale V by R plus S this is the same as just scaling V by R and then scaling V by S and adding the answer together that works too and similarly if I scale with some vectors that's the same as scaling independently I really hadn't planned to write down all of these properties that give us a vector space I sort of got lost anyway so we have that so we have these vectors I'll finish them by factors I guess I neglected to put subtraction if I want to subtract one from the other then I go it's the other diagonal that's the difference now it's a little hard to see let's see why is it now so if I put negative this is V minus W because I put so this is V minus and the other diagonal is the difference what you can see by just taking W flipping it around sticking it there if I take this W with minus W here with that on the there you're lost? Me too yes it doesn't make any sense so this is not defined for us we can't multiply vectors we will have two operations and I'm getting to one right now that work like multiplication but they are not multiplication in the same way that 2 times 3 equals 6 is multiplication they just share many properties of multiplication and we use the same symbols but they do not mean multiplication if you think about it I have an arrow and I have another arrow and I want to multiply that what does that mean? it doesn't mean anything I don't know how to multiply arrows I know how to add them because I just picked this one up and put it on the end that's added but I don't have a good way to make good sense of multiplying I mean there are operations that look kind of like multiplication and we'll talk about them very soon that give us either another arrow or give us another but yeah this doesn't make sense so in terms of arrows everything I wrote here makes sense adding them means stick one on the other scaling it means stretch it by that factor so if we just think of it as little arrows this makes sense and that doesn't but now that I said we can't do it I'm going to do it but it's not the same as what two times three means so one thing you might try to do suppose I have a vector v which is 1, 2, 3 and I have a vector w which is 1 minus 4 5 you might say well let's multiply this stuff together so let's just take this times this, this times this this times this well then let's add them together just for fun so this actually means something it's kind of a way to multiply vectors but it's a number right this is 1 minus 8 plus 15 0 8 0 8 it's one of those numbers ok so this is a number and it's a number that has something to do with me and w I can define this it makes sense and if you just write this down and I'm just going to call this v dot w because it's kind of like multiplying and maybe I should use a little x so this is something called the dot product multiply the coordinates together and add them up but it means something in terms of the vectors and I want to, so let's look at a few properties of this guy before I try and interpret what it means well if I have a vector v so certainly it's defined for any two vectors I write down so it works for every two vectors and if I have a v and I dot product it with a w it gives me a number it will give me exactly the same number as if I do live in the other order the order doesn't matter because multiplication is commutative and different and addition is commutative so if I change the order around it doesn't matter I get the same answer so this is symmetric what's another property that this has well let me tell you because I have it in mind if I take any vector and I multiply it by itself in this way what can I say about it this is like v squared but it's not v squared it's always positive except for the zero vector so if v is the non-zero vector then the dot product with itself is positive and the zero vector the dot product with itself is zero so the only vector where the answer is zero by the dot product with itself is the zero vector now that's a useful thing actually because well let me finish this we also have kind of a distributive business if I take if I take so notice that this doesn't make sense v dot w dot z this doesn't make sense because this is a number and the dot product is only defined on pairs of vectors so this is meaningless yeah I think vector v times vector of w is a number times another vector should be another vector that's right these are dots so dot means take two vectors give me a number I can't take a number and dot it with a number so this dot so this means something if this isn't this this becomes a scale which then I can scale it but this is not the same as this so this kind of doesn't make sense once I combine two vectors I don't have another vector to combine them with I don't have a scale so I can make sense of v dot w times z that makes sense that means stretch z by the dot product of v and w but I can't make sense of v dot w dot z so it's important to remember what are the domains of these operators these dot things these vectors and it gives you a number ok but I guess coming back to this for a minute the way I defined it this has an inherent geometric meaning I have some vector x and I dot it with x this gives me a number let's call it a gives me a number what is the relationship between number and this vector a I mean vector x it's the length of x right this is the length of x let's see why that is so I have some vector x which has coordinates x1 x2 let's just stop it and if I think about let's do it in 2 it works in any number but let's just do it in 2 x1, x2 now that means that this direction here is x1 and this length here is x2 and by the Pythagorean theorem this length is the square root of x1 x2 x1 I don't know what I'm doing x1 squared plus x2 Pythagorean theorem which is well if I take x1 x2 and I dot it with x1, x2 I get x1 squared plus x2 squared so this is x dot into the x squared so we can pull out the length of the vector by this dot product thing and this statement is just a way of saying every vector has a length except the zero vector has a positive length and the zero vector has zero length so this statement says just what I said every vector has a non-zero length except the zero vector this is really the square of the length but ok again let's and it should be apparent to you that if we're in free space you can use the distance formula and we'll get the same answer if we're in 12 space use the distance formula for 12 dimensional space but this is sort of a convenient way to encapsulate the distance formula to length of the vector and another property suppose I have two vectors playing around I have y here I have y and here I have some vector x and so let's do the simplest case or the perpendicular let's start with that case first so this is the right angle the way I drew it but let's start with this case and now what will x dot y be? well so if you know what the dot product is you know it's zero but let's see that it's going to be zero so how can we see that? how many people already know about the dot product? so like I'm wasting all this time explaining how you think well those of you that don't you listen you guys go to sleep okay so let's calculate x dot y in this case and actually I don't want to do that okay so I plan that this is zero and well in this case it's sort of particularly easy because I drew it in two dimensions I can write this as I put my x on the y axis and my y on the x axis so this is the vector y zero and this is the vector of zero x and so when I calculate their dot product x they have little hats x dot y is going to be zero times y plus it's not a vector zero times y okay so now in higher dimensions it's still going to work the same way but let's bend it a little bit so if I have some angle between them well so this vector here is the vector x minus y and I want to somehow relate x minus y to x and y and theta pretend for a minute that we're in a geometry class we don't have vectors that's not a right angle it looks like one this is some angle theta this is some a this is some b this is some c does anybody remember something about the relationship between a, b and c? the love of time so what does it say? be sure it was c squared equals b squared plus j squared minus which would really be cosine and z and that would be right so this is the law of cosines and it relates to the lengths of the sides of a non-right triangle or a right triangle because in a right triangle theta is 90 degrees so the cosine of theta is zero so this term is gone and we just get the Pythagorean theorem as a special case so here we have this is my c and then I get x squared plus y squared c squared and it's the same because 2xy cosine of theta, theta is 90 degrees so that term is gone so we have this and I want to somehow relate this to the dot product to pull out something about theta so in this picture where a is x a is y and c is x minus y let's just rewrite the law of cosines using those vectors well they're not vectors really I'm just thinking about the lengths I'm sorry c should be y probably no c is x minus y in this picture yeah did I mess up my okay so I just want to plug this in where this is c this is a this is b so it should be true just from the picture it's a triangle so it's a triangle so it's still true that x minus y square should be equal to x squared I've permuted the order I'm sorry but things are commuted in the right way so I'm lucky minus 2ab sorry 2xy cosine theta so that should tell me something and I want to try and pull out something about theta in terms of the length of x and the length of y and maybe the dot product or something like that okay so now I'm going to write x minus y squared in a different way I'm sorry I'm going to write it this is a vector it's the length of this vector so the square of its length is just the dot product of the vector x minus y that's a vector and then when I started writing down the rules and didn't finish them I'm going to use them so I should have written them down but you bet so we have a kind of a distributed law here this is the same thing I can say well that's going to be x dotted with x minus y that's okay minus y dotted x minus y those are pairs of vectors so does everybody believe that that works yes you know just writing symbols so whatever okay and then I can play it again x dotted with x two vectors minus x dotted with y minus y dotted with x minus well it's plus because I have minus minus y dotted with y and then I can rewrite this in terms of length this is the length of x square this is I'm going to reorder plus the length of y square and then this is minus two x dotted with y okay well now comparing this to the law of cosines that tells me this is a vector which is the opposite side of this triangle this is the length of the opposite side of the triangle this is the length of one side this is the length of the other side and this is something so looking comparing this to the law of cosines I see that it must be true there's no other way that this and this are equal but they are because those are the same thing is if x y cosine theta is exactly x dot y there's no other way that this will be true so I've just calculated the cosine the angle between them is the dot product divided by the product of the length this is a very important calculation of the pool and so the dot product which more formally is called the inner product but the inner product of two vectors tells us the cosine of the angle between them doesn't quite tell us the angle close but in particular you get that when is the cosine zero the cosine is zero exactly when these two are orthogonal so we also get that x dot y is zero exactly when x is perpendicular to y so this dot product thing is a very useful object it allows us to measure angles and lengths and in particular tell when two things are orthogonal and notice that all of this calculation didn't use anything about coordinates it just used properties of vectors so this this calculation although I sort of drew the picture in the plane even if we are in three space or seven space there is a plane that goes through two vectors so there is a well defined triangle so all of this stuff works in any dimension it didn't do anything bad it doesn't generalize to any number of dimensions so we have a well defined angle between two vectors and we know how to calculate it we can tell when two vectors are orthogonal no matter what no matter how many dimensions we are in and that's a really useful thing okay so I have a way to do lengths so if I have a way to calculate lengths this means I have a way to define length one so the dot product measures angles and gives lengths and in fact so I'll just say these words so we can generalize instead of using a dot product more generally we have something called an inner product which in the Euclidean in the standard Euclidean space that we are working with in this case standard Euclidean space that we are working with in this class this inner product is the dot product but we can make sense of vectors that aren't arrows in three dimensional space vectors are a more general concept of polynomials of vector so if you have an inner product you have a way to measure angles and lengths so there's this notion you might have heard the word Riemannian geometry or Riemannian metric anybody heard this word before okay sorry R I E M A N N you've seen his name before yes his name wasn't E yet his name was Bernard so Riemannian geometry this means we have an inner product so we can generalize this notion of flat spaces to curved spaces and the inner product or the way we measure angles between things tells us how the geometry works how the space works the geometry that works in the normal sense this is not a course on Riemannian geometry although Riemannian geometry is a generalization of a lot of what we're doing but if we generalize this notion of inner product we get a more general kind of geometry than just flat geometry and so there's a lot of people at Stony Brook that study Riemannian geometry not Riemannian geometry okay so that was just I don't know me saying stuff okay so a unit vector sometimes we might want to just think about directions and not worry about lengths so if you think about like a compass point this is supposed to be a compass so we have a compass here and we just have an arrow that moves around and gives us a direction or in space or whatever and we don't care how long the arrow is so we might as well always make it 1 so we can think of a unit vector it always has a direction and it's very useful unit vectors have this property that they don't when you use them they don't sort of stretch things so just to make a unit vector set the zero vector and we just divide it by its length this gives us the unit vector u which is just sort of a pure direction because the length is 1 so they don't care about the length so a lot of the formulas become a little simpler if we're dealing with unit vectors but I want to mention the idea of unit vector is a lot of times the word of unit vector comes up let's just take the vector divided by its length it's 1 so the length of u is 1 3 minutes so another thing that we often want to do with vectors is you might want to cast the shadow of one vector on another you might want to take the component of one vector in a certain direction so I might have a vector here v if I look in this direction how long is the shadow of v here so this is maybe let's call it w because I don't necessarily want to specialize to a unit vector w so I want to know how long is the shadow or the projection in the w direction so this is a question that is very useful maybe it's obvious at this point why it's useful but it is useful for doing things like calculating distances between two lines or distances between planes or stuff related to calculated distance projecting one thing onto another is a useful thing so I claim so when we talk about the distance between a point and a line we always mean the minimal distance so that's the length of the perpendicular so this is the distance between the point and I want this projection which is the other component so how would we do this yeah I think it's like v dot w over w dot w times w is this because you just thinking about it or is this because it's a formula that you saw before you need to if you do v dot w you get absolute value of v times the absolute value of w and then we know in the end no matter what we're going to be multiplying by the vector w right and then no matter what we know that we're going to be multiplying by the vector w in the end in the roman direction so you'd write that up too like vector w and then we need to divide by w okay so so we have this angle here and certainly v dot w is the length of v, length of w cosine of the angle between them and but we also want a vector so and I want to think of w as a unit vector and now I just look into w direction so that's what you said this is not v dot w anymore sorry so this is the projection of this thing onto that and since I'm out of time and they're going to come in here let's stop here, I'll pick up with this on Wednesday I think there's some of this on the homework so there is a homework it's on the website


It's Ellie Hart from TC Helicon and in this video we are going to be talking about presets with the Perform VK. So the Perform VK comes with three different preset settings. What's a preset? Well, it's basically a collection of effects that you've pre-assigned to that preset. How many times did I say pre in there? It doesn't matter. So, for example, if I want a set of effects for all of my fast songs, I can create a preset for fast songs. I can do a second preset for slow songs and then a third preset for any weird and wonderful other combination of effects that you think is appropriate for your set. So what you do is you have your first preset highlighted and you can go in here and any edits you do to these effects is going to be saved to that preset. When you go to the next one you'll have any edits again saved to that one. So, for example, I have this red reverb, hey, hey, and then for echo I have, hey, check, check, there. And this is kind of my fast song. Who can't fight the temptation when you feel the vibration? And that gives me a nice short reverb with a really fun kind of live rock and echo. Okay, preset two. I've kind of made this my slow preset. So I've got a, it's really nice, long reverb. The nice thing about these presets is that if you choose a harmony voicing under this preset it will stay there and it won't change. So for my slow one I've got harmony low and lower selected and that's from my cover of Let It Be. So for example, if I selected preset one and I did that song it would sound different. It doesn't necessarily sound bad, it's just not the sound I was going for. I wanted that, those effects from preset number two. So preset number three, I've got set up right now to work for my song Hide and Seek by Imogen Heap. So I've got the harmonies pre-programmed to do my MIDI notes and reverb and delay all programmed. So all I need to do if I want to do this song is I hit my octave button here on my keyboard and I turn the volume down. I'm all ready to go there. It also takes the harmony voice level, volume level. You can individually determine that for each preset. So basically anything you do in a preset level, like any edits you make to your effects is going to save to that preset. So that's really a lovely function that we have there. So we also have several different presets and even different effects styles available that you can import into your PerformVK using our phone app called the PerformVK. So just go to your iOS or Android App Store, look for PerformVK and this app will show up. Download that, it's free. The first screen that will come up is this beautiful photo of the PerformVK. There's a search bar, so if you already know the name of the preset or the effect that you're looking for, you can use that. You can also hit the genre button. It actually gives you a list of different genres that you can choose from, that have lots of different presets within there, or just the effects themselves, hard tune, double, echo, reverb, megaphone, extreme, and there's some character presets and effects in there as well. We also have a whole bunch of presets that we've created based on popular songs and popular artists, so you can have a peruse through there. And then once you like something, we also have a favorites menu so you can save these different presets to your favorites. So let's say I want to get this extreme, we'll just do something extreme because why not? These are actually, these are different individual effects, I believe. So I changed my mind. We're going to get a preset. So I'm going to go to the songs and choose Aki Breiki Heart. If you want to make that a favorite, you just hit that star and it's going to show up in your favorites menu. So Aki Breiki Heart, I can see the effects that are in it. There is reverb and there's some harmony. So there's two different ways to get these effects into your Performed VK. One is by beaming it in and the other is by using USB. So I'm going to show you how to beam it in. What you're going to do is you're going to make sure that the volume on your phone is turned on. So I've got my little volume button on the side. I am going to put this next to this little microphone that's on here and I'm going to hit beam. And it says wait till the sound stops. There we go. It makes a really fun sound. It feels like you're communicating to somebody in space, like breaker, breaker, 1, 9, brrrr. Anyways, it's beamed in there. So that just beamed in to preset one and now I've got, I've got that in there. You'll also notice that when I hit it again, it went blue. So that's called a hit. All of our presets that we have within the app have hits and hit allows you to trigger something different in the middle of the song. So when this preset you hit hit, it triggers harmony and it lets go of the delay. So maybe in the chorus, they don't have delay in the song. So don't break my heart. Same thing. You need to give it some chord information. Don't break my heart, make it break your heart, I just don't think you'd understand. Okay so if you want to beam in a preset in the middle of a set but you don't want that sound to happen, the only option is to use your USB and a camera connector kit. So this goes into here. This goes into your phone and this end goes into the back of the Perform VK. I'm using it for something else right now so I'm not going to do that but that's your option and yeah, very simple. You just choose your thing and you hit beam still but it'll go through this instead of going through the, through the mic and making that noise. So that's pretty simple. So let's go back to this preset that we did beam in there and I've got hit. Now hit basically means if you have a chorus coming and you want to change the effects with one button you can change multiple effects. So say green for preset one is just reverb but in the chorus I want harmony and delay to start. Instead of hitting both those buttons you can hit the one again, it'll turn blue and we'll have harmony and echo come on. So again, say we're in the verse, we're in preset number one, we've just got reverb. Now the chorus comes, I want harmony and echo to go, just hit the hit. Now blue is hit activated and those two came on. So if that's what comes with the preset but I'm like, I'm an artist and I have my own opinion about how I want this to sound and I don't want delay or echo to come on. While I'm in that hit state, the button's blue, I can tap echo, deactivate it and now it will not come on when you hit. So I've gone back to my normal screen with green, hit, you'll notice echo did not come on that way. So super simple. The other thing if you want to edit any of these effects from the preset it's the same pretty much as editing effects from before, hold down reverb but you'll notice that the first edit screen it takes me to is that white light blue color. So that's the beamed in or imported in preset or rather effect. It's always going to go to that field, that reverb style, same with echo. Hold that down to edit, it's going to automatically take you to the beamed in or imported in effect and that's that white color. It can be a little bit confusing because when it times out, you'll see it's not pulsing anymore. It's kind of that this exact same white light blue color. That's remember always going to be the harmony voice volume and you can tell because when I start moving it, the harmony starts pulsing. So again, if I hold down echo to edit it and I'm moving it around, it's the echo that's pulsing not the harmony. It times out pretty quick. There we go timed out back to this white but nothing's pulsing as soon as I move that control knob. It's the harmony that's pulsing. I hope that helps clarify that a bit. So that's how you edit, that's how you import presets and that's all the things you need to know about presets and hits. So I hope that was really helpful. If you have any further questions, throw them in the comment section below. Don't forget to like the video and if you feel so inclined, please subscribe to our channel.


Everybody force to your rip in the sports card almanac multi-sport personal edition box Volume three here. This is box number 54. This is rich K on number 54. Good luck rich 54 Yeah, it's been pretty fun hunches pretty fun for a Erlacher So I really think I've left 54 some other sports. Oh good time to get this one. I tell you Yeah gold frame transcendent collection auto Aaron judge five of 15 Heaven and all time season right now 2021 Transcendent baseball Aaron judge auto five of 15 Facebook's all weird. Oh No, it's just a glare. No, it looks good. That's all 2021 was just transcendent. We'll haul it from now. Just regular transcendent. That's awesome man Gold frame judge five of 15. Congrats rich. Like I said, great time to get this. We'll get it out too shortly


The true face of Israel notes from the edge of the narrative matrix Israeli Defense Minister Yov Galant has announced a complete siege of all of Gaza in response to the Hamas attack on Saturday Justifying the deliberate targeting of civilians with siege warfare by the claim that Israel is at war with human animals. I Have ordered a complete siege on the Gaza Strip. There will be no electricity. No food. No fuel Everything is closed. Galant said Monday adding we are fighting human animals and we are acting accordingly So there you have it folks. They're attacking civilians, which is fine because they're not actually attacking human beings There it is The true face of Israel naked and unadorned in the cold light of day The mask is completely off now Israel apologists get mad at me for criticizing Israel when Israeli officials are saying much worse things with their own mouths About what Israel is than I ever have a Lot of evidence has come out casting doubt on the Wall Street Journal claim that Iran coordinated the Hamas attack along with Hezbollah Moon of Alabama has a nice write-up of the glaring plot holes in the story Including the former employer of its main reporter saying he fired her from Reuters in 2008 because she has a history of dishonesty and inventing stories Israel has also said it has no evidence Iran was involved Suspicions that the Saturday attack was allowed to happen have been given more weight by an Associated press report which cites an anonymous Egyptian intelligence official saying Egypt had warned Israel Something big was in the works in advance of the attacks We have warned them an explosion of the situation is coming and very soon it would be big But they underestimated such warnings the official said Multiple US officials have deleted tweets calling for ceasefire negotiations to end the conflict Secretary of State Tony Blinken and the Twitter account for the US Office of Palestinian Affairs Both removed posts from the platform the former calling for a ceasefire mediation from Turkey and the latter urging all Sides to refrain from violence and retaliatory attacks The exact reasons for the deletion are unknown, but the general reasons are politically obvious prominent figures have been referring to the Saturday attacks as Israel's 9-11 Which should set off immediate alarm bells in everyone's head The most significant thing that happened on September 11th, 2001 was not the 3,000 deaths from the attacks themselves But the dawn of an age of Western interventionism and military expansionism that would go on to kill orders of magnitude more people than died on 9-11 That's what you should think about when people begin comparing a new event to 9-11 Not to the event itself, but to the mountains of unwise decisions of far greater consequence that were made in its wake That should be an automatic association in our minds We should become more skeptical and oppositional toward the warmongering agendas of our government and its allies when we hear things compared to 9-11 Not less It's hilarious that it's 2023 and people still think calling you a terrorist defender in an anti-Semite will stop you from criticizing the abuses of a nation which all leading human rights organizations have now labeled an apartheid state I built a new house There were people living there where I wanted to build it. So I just started building it on top of them They tried to stop me.  I'm trying to build Or b build somewhere else


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Hello and welcome to the session. In this session we are going to discuss how to convert polar form into j operator form and voice versa. In complex numbers we have studied that a complex number is of the form a plus iv where a is the real part and v is the imaginary part. We also know that i is equal to under root of minus 1 is called the imaginary unit. The j operator is same as i that is the imaginary unit. The people in science and technology use j for the imaginary unit thus in technical mathematics the imaginary unit is represented by the symbol j where j is equal to under root of minus 1 and it is known as the j operator. So a complex number is of the form a plus jb where a is the real part and v is the imaginary part. Thus we shall make a note of the following things. j is used instead of i in technical mathematics, i is used in pure mathematics and j is used in electronics. Thus a plus jb is same as a plus iv. Also the representation of a plus jb on argon plane is same as that of a plus iv. The rectangular form of a complex number is given by a plus jb. For example 2 plus j4 is a complex number with real part 2 and imaginary part 4. Now let us discuss the representation of a complex number on argon plane. We know that the representation of a plus jb on the argon plane is same as that of a plus iv. So here on the argon plane x axis is the real axis and y axis is the imaginary axis. The complex number is represented by the ordered pair ab on the complex plane. It is drawn in the form op where op is equal to a plus jb. Now next we shall discuss the polar form of a complex number a plus jb. The polar form of a plus jb is same as the polar form of a plus iv. We can write a plus jb in polar form using trigonometric ratios. For this we put a equal to r cos theta and b equal to r sin theta. So a plus jb is equal to r cos theta plus jr sin theta. So here if we take r common this is equal to r into cos theta plus j sin theta the whole where r is the modulus theta is the amplitude. Also r is equal to under root of a square plus b square where r is strictly greater than 0 and theta is equal to tan inverse of b by a. Another method of representing polar form is r angle theta which means same as r into cos theta plus j sin theta the whole. So here on the argon plane the horizontal axis is the real axis, the vertical axis is the imaginary axis which represents the j operator. The vector represents the complex number a plus jb which is r angle theta or r into cos theta plus j sin theta the whole in the polar form. Now we shall discuss how to convert polar form into j operator form and vice versa. We have already discussed conversion of polar form of a complex number of the form a plus ib under the topic moron complex numbers. We will follow the same procedure and steps to convert complex number of form a plus jb into polar form and vice versa. Let us consider the following example write 1 plus j root 3 in polar form. On comparing it with a plus jb form we get a equal to 1 and b equal to root 3. So we first find modulus r for this. Since r is equal to under root of a square plus b square where r is strictly greater than 0 on putting values of a and b r is equal to under root of 1 square plus root 3 square r is strictly greater than 0. So r is equal to under root of 1 plus 3 which is equal to under root of 4 which is equal to plus minus 2. But we will take r equal to 2 as r is strictly greater than 0. Now next we shall calculate the amplitude theta. We know theta is equal to tan inverse b by a. Again putting values of a and b theta will be equal to tan inverse root 3 upon 1 which is equal to tan inverse root 3. And as we know tan inverse root 3 is equal to 60 degrees. So theta is equal to 60 degrees. So we have found that the modulus r is equal to 2 and the amplitude theta is equal to 60 degrees. So the polar form of 1 plus j root 3 is equal to r into cos theta plus j sin theta the whole. When we put the values of r and theta the polar form of 1 plus j root 3 is equal to 2 into cos 60 degrees plus j sin 60 degrees the whole. Now 60 degrees is equal to pi by 3 in radians. On writing angles in radians we get radians 1 plus j root 3 equal to 2 into cos pi by 3 plus j sin pi by 3 the whole or it can also be written as 2 angles 60 degrees. Now we see how to convert polar form in j operator form. For this let us consider the example convert 3 into cos 120 degrees plus j sin 120 degrees in j operator form. We know a plus jb is equal to r into cos theta plus j sin theta the whole. So on comparing r into cos theta plus j sin theta the whole with 3 into cos 120 degrees plus j sin 120 degrees the whole we get r is equal to 3 and theta is equal to 120 degrees. Now since a is equal to r cos theta and b is equal to r sin theta putting the values of r and theta we get a equal to 3 cos 120 degrees and b is equal to 3 sin 120 degrees. Now since cos 120 degrees is equal to minus half and sin 120 degrees is equal to root 3 by 2 we get a is equal to 3 into minus half which is equal to minus 3 by 2 and b is equal to 3 into root 3 by 2 which is equal to 3 root 3 by 2. So a plus jb is equal to minus 3 by 2 plus j into 3 root 3 by 2 which is the required form. So in this session we discussed the j operator form and how to convert polar form into j operator form and vice versa. This completes our session. Hope you enjoyed the session.


Peers in the class 3A quarter finals playing Proctor at Brayden on Saturday opening pioneers drive Reese Castner powers in from a few yards out for the score. Peers on the board first then Moore on their next drive. Capsner again hitting all sorts of buttons on the Xbox controller. Six more extra point makes it a two score advantage and for as good as the offense was the defense just as good. The pitch to the Proctor running back goes nowhere. A swarm of pioneered offenders there for the stopping back to Castner. This went a little further even too fast for the camera. Brakes free of the open and a date with the end zone. All Peers in this one 42-13. They have Jordan the semi-final Saturday afternoon at US Bank Stadium. If you've enjoyed this segment of Lakeland news please consider making a tax deductible contribution to Lakeland Public Television.