What is up you guys, welcome back to another one. If you are new to the channel, I'm Gold Pony, I do new car truck SUV reviews on YouTube. And today we are in the new 2020 Volvo XC40, courtesy of younger Volvo in Hagerstown, Maryland for more information on their inventory. Please feel free to check out the link in the description box below. Wanted to check out this one, this is Volvo's entry level SUV into the Volvo family. My IHS top safety pick, few new changes for 2020, all of these reasons being why I wanted to check this one out today. So what do you guys say? Let's just go ahead and jump right into it. And as always, let's start with pricing. And so as you can imagine, there are several different trip levels for the 2020 XC40. First one being the T4 front wheel drive momentum, starting at $33,700. T4 front wheel drive are designed for $39,950. T4 front wheel drive inscription for $40,450. Then you have the T5 all wheel drive momentum, actually the one we have today, starting at $35,700. T5 all wheel drive are designed for $41,950. And the T5 all wheel drive inscription for $42,450. And so as you can imagine, there are actually two different engine configurations, one of them belonging to the T4, the other belonging to the T5. So I'll start with the T4, although that's not the one we have today. The T4 is powered by a two liter turbo charged inline four cylinder, putting out 187 horsepower at 4,700 RPM, 221 pound feet of torque available at 4,000 RPM. Power sent to the front wheels, like I was saying through an eight speed automatic, zero to 60 on that one, approximately 7.8 seconds. MPG numbers, 23 in the city, 33 highway. Premium fuel is recommended, however, regular fuel is okay. But so then there is the other engine configuration belonging to the T5 trim levels and the one we have today. This one is going to be a two liter turbo charged inline four cylinder once again, but this time putting out 248 horsepower, 5,500 RPM, 258 pound feet of torque available at 4,800 RPM. Power sent to all wheels. This is the all wheel drive configuration engine with the eight speed automatic once again, zero to 60, approximately 6.7 seconds. We will give it a little acceleration later. So we'll see what that actually feels like. MPG numbers coming in at 22 in the city, 30 highway again, taking premium, unleaded fuel. And so before we do that acceleration, I did want to mention there are some drive modes, of course, for the XC40. Drive mode button is going to be located just below the infotainment screen. It is one of the seven or eight buttons included in the XC40, a very minimalist design, which I love. But anyways, drive modes are going to include comfort, eco, dynamic and off road, adjusting things like the shift points, steering sensitivity, braking, suspension and the climate control settings actually, as well, so quite a bit being adjusted with those driving modes, it's kind of cool. But anyways, I do currently have it in that dynamic driving mode, which is the fun one, essentially, holding the RPMs a little higher. So we are prepared now to do that acceleration. What do you guys say? Let's find a straightaway and let's see how quickly we can get the new 2020 Volvo XC40 T5 up to speed. All right, you guys pulling out onto the road. Let's see, whoa, it's the ghost. Wow, okay. Certainly no issues with marching on to the highway. I think what surprised me the most was the initial punch. Like a lot of times you think with turbocharged engines, there's just going to be a little bit of turbo lag, a little bit of delay when you first hit the gas, but there was none, there was no turbo lag whatsoever with the XC40, which is brilliant. Instant acceleration instantly merges you onto the highway. No issues with acceleration whatsoever, at least with the T5, it can vouch for the T4 because I'm not driving it, but the T5 is wonderful. But anyways, one thing in addition to that, if you're wondering about paddle shifters, you can get them, but they do come with the R line trim level only, so therefore we're not going to have them today. But anyways, to go along with that acceleration, as always, braking is equally important. So upfront, you will find 13.6 inch ventilated front discs in the back, 11.9 inch solid rear discs. By the way, that braking setup goes for either engine configuration. It's going to be the same either way. Braking feels been perfectly fine for me today, so absolutely no issues there. It feels wonderful. Touching on suspension and handling, upfront you're going to get double wishbone type front suspension in the back, independent multi-link rear suspension. And I did want to mention there is an optional package. It's called the 4C suspension. That goes for $1,000, but that essentially is just your adaptive damping suspension, which is usually an option I'd like to recommend because that essentially does give you the best at both worlds because it monitors each shock absorber individually. So it's not only going to give you a smoother ride, but when you're going around the turns a little faster, it's going to tighten up that suspension for better handling as well. So therefore, that's definitely an option I typically do like to recommend. So anyways, ultimately when it comes to ride quality, it's been perfectly fine for me today, although in Maryland we do have some pretty smooth roads here, I will say, compared to where I live in Pennsylvania. So perfectly fine for me there. As far as cabin noise goes, you guys could probably tell there isn't a whole lot coming into the cabin here. Even cruising at 55 miles per hour right now, there isn't a whole lot of cabin noise. So Volvo did a pretty good job with the XC40 when it comes to cabin noise. I'm kind of impressed there actually.  But honestly, with the XC40 even going over a bridge right now, it's really not that bad. So perfectly fine for me. Steering feel is brilliant, not the heaviest thing in the world like my Mustang, but steering feel is plenty nice. Definitely a nice way to it, at least in that sport driving mode that I currently have it in. So no complaints from me there either. And when it comes to visibility, I actually can see perfectly fine out the back. So actually visibility is one of the better ones I've seen lately. That's really good. Absolutely no issues whatsoever with visibility. So I think it's due to the shape. A lot of times with boxer SUVs, when it comes to the back of it at least, you are gonna get better visibility. And that is certainly the case in the XC40. So wonderful visibility in this one. And so to go along with that included with visibility really, rain sensing windshield wipers do also come standard on the XC40 regardless of which trim level you go with. That's always nice. Essentially what that means is whenever the XC40 detect any kind of mist or rainfall, it's gonna automatically turn on as windshield wipers for you. So it's kind of like automatic headlights. It's just one less thing you have to worry about when you're driving. So that's a big old plus there too. But that about rounds out the performance segment of this review you guys. Let's now go ahead and take a look at the exterior of this brand new 2020 Volvo XC40. All right, so here she is you guys. The 2020 Volvo XC40 finished in Fusion Red. In case anybody was curious about that exterior color name. Let's go ahead and start up front on this one. LED headlights with Thor's Hammer LED daytime running lights in typical Volvo fashion coming standard on this one. That does come with the automatic feature by the way for both low and high beams. A lot of times it'll only come standard with the low beams but on the XC40, high beams are included. So that's pretty cool. Full LED headlights with active bending come part of the advanced package. That's a $1400 option by the way. But active bending essentially is when you're going around the bend at night those headlights will swivel based on your steering angle. Better help illuminating what is around that bend so you're less likely to hit a deer or any other objects basically. LED fog lights also coming standard. Aluminum accents on the bottom portion of that front bumper that is going to come standard on this one. And the grille is actually going to differ amongst the trim levels. For example, if you go with the momentum trim level you will get black vertical slats within that front grille. If you were to go with the inscription you're going to get chrome vertical slats and the R design is going to give you a black block style front grille. So they are going to differ substantially amongst the trim levels. I guess that's a good way of telling the difference between the trim levels I suppose. So let's take a look at the front grille. But anyways, let's go ahead and swing around to the side on this one. Silver roof rails coming with the momentum and the inscription trims. Black roof rails with the R design of course. Rear privacy glass coming standard across the board. And when it comes to the window surrounds you will get black window surrounds with the momentum and the R line. However, you will get chrome belt line molding if you were to go with the inscription trim level. Let's take a look at the side mirrors. They are body colored power adjustable side mirrors for the momentum and inscription trim levels. Gloss black side mirrors with the R design and either way they will be heated with integrated turd signals across the board. So that's always nice. Let's take a look down to the wheel setup. 18 inch five spoke silver alloys coming with the momentum. 18 inch six spoke diamond cut alloys with the inscription. And lastly 19 inch double five spoke alloys if you were to go with the R design of the XC40. But now let's go ahead and make our way to the back of the XC40. I can already see it up top. Shark fin antenna coming standard across the board. Respoiler with an integrated brake light coming standard just below that rear window wiper. Also that Volvo lettering spelled out horizontally. I've always liked that look. Trim level badging back there but just below it all when it comes to the exhaust setup it is gonna differ quite substantially amongst the trim levels. Let me see if I can even show you guys down here. Dual exhaust outlets hidden if you were to go with the momentum. So obviously that's what you're looking at right now. However, dual exhaust outlets integrated into the rear bumper with chrome tips. If you were to go with R design or inscription but nonetheless, as always you guys know what we have to do next here is that exhaust clip. And so now since we are around back when it comes to opening that rear lift gate it is a power lift gate by the way. I did want to start by mentioning that it can be a hands free lift gate if you go with the premium package it goes for $1,900. But if not all of your buttons on the key fob there's actually a button to pop that rear lift gate on the side. All the buttons are on the side of the key fob and there's also a button on the lift gate itself. You just simply press it. You don't have to actually pull up on it. Just press it. It's gonna open up itself and then you press another button to actually close the lift gate. It's pretty convenient back there. Once opened up behind that second row cargo capacity comes in at 20.7 cubic feet. If that was not enough space however you can fold that second row down by bumping it up to 47.2 cubic feet. And by the way that is a 60-40 split and you can also find some cargo lighting back there. There is some in-floor storage and a 12 volt power outlet actually as well. The maker way up to the rear leg room that is gonna come in at 36.1 inches. So for reference I'm even six feet tall. This is how much space I have back there. Rear passengers also have a rear center armrest with cup holders. There is a 12 volt power outlet. Little bit of storage space. And you have some coat hooks located on the ceiling if there weren't any rear passengers back there as well. But now make your way up to the front seats. Eight-way power adjustable driver seat with power lumbar comes standard. And that does come with memory settings by the way for up to two different drivers. That can be found on the driver side door there. But now one of the main changes for the 2020 XC40 is going to be the cloth leather combination for the momentum. That is gonna be a 2020 specific change right there. So although we do have optional full leather seating today so I can't really show it to you guys but that is probably the main change for 2020 XC40. Although I will say if you go with the R design or the inscription trims that is gonna give you full leather seating as well. Heated front seats are available for $750. That comes with a heated steering wheel as well. You had another option that we do indeed have here today. Overall the seats have been plenty comfortable for me. So certainly no issues there. And of course you got the flag of Sweden found on the side of the passenger side seat there. So I always liked that they put their flag there. Kinda makes this car a little bit extra special driving it but anyways. Now let's take a look at the steering wheel. There's tilt and telescoping. It is leather wrapped across the board. And if you wanted a heated steering wheel like I was saying that is optional but it is available for you there if you wanted it. And take a look at the startup. Let me first start by showing you guys the key here. You do have your Volvo logo on the one side. Absolutely nothing on the other. And again all of your buttons are actually located on the side of the key including lock. Unlock the button to pop the rear hatch. And on the other side you have your panic button but all I'm going to do is all keyless entry. Just simply leave the key in your pocket. Put your foot on the brake. There is an engine start button located just by the driver's right knee there. And I should mention before we get to the gauges this transmission setup is a little bit different from some of the other manufacturers. You simply just slide the shifter back. That puts you in neutral. Slide it back again. That puts you in drive. Slide it forward. That puts you in reverse. And park is actually going to be a button just behind the shifter there. So then you got your parking brake button of course as well so don't get them confused. Anyways, let's now get to the gauges here. 12.3 inch digital gauge cluster coming standard. Absolutely freaking love that. Looks absolutely amazing up there. To control what is on that digital display there are a couple of buttons on the actual steering wheel on the right side. Need to of course check out different things using that but how many miles you have left until you hit empty it gives you your speed limit recognition technology up there as well. Digital speedometer, tachometers on your right time of the day, outside temperature your basics basically. But it looks absolutely wonderful up there. I will say that. And it will actually adjust ever so slightly depending upon the driving mode that you put it in. Like for instance, off-road is going to give you a compass at the bottom. I found that pretty cool. And the eco driving mode is also going to give you some green hues as well. So overall very cool digital gauge setup. Let's take a look at overall interior quality though. Panoramic moon roof that we have today that goes for $1,475. If you wanted that, charcoal headliner goes for $200 universal garage door openers. You can get them with the premium package but that can be found underneath the frameless rear view mirror. I always like frameless rear view mirrors but it's up to three different garage doors on the bottom of that there. Wireless phone charger though also coming standard with that premium package. There is also a convenience package I wanted to mention now. $600 but that gives you a storage box under the driver seat cushion. That's kind of unheard of. You usually don't find something like that but I think that's super cool. Also it gives you automatic climate control and a power front passenger seat as well and some other things. But you can also find driftwood inlays if you were to go with the inscription trim level. So therefore we don't have them today but we do have this silver trim inlay which is actually kind of cool. It feels pretty heavy duty. Has a nice design to it and overall a very simplistic design to the XC40. And really that's kind of their thing, simplicity. So you have a total of like eight buttons found just below the tech display and that's really where you can control everything, the tech display. But just below that 12-volt power outlet USB charging ports times two it looks like there. Then you got your wireless phone charger, a little bit of storage to the side there. Also a little more storage that opens up right in front of that. That's pretty cool. Dual cupholders, you got a little more storage just behind that. And within the center armrest you have plenty more storage there as well. So overall very nice interior quality, very simplistic interior quality which I actually like. But now let's make our way to the tech display. Keep mentioning nine inch color touchscreen display comes standard on this one. Bluetooth and audio streaming also standard Android Auto Apple CarPlay. Factory navigation system is available though you don't need it if you have a smartphone because the Android Auto Apple CarPlay but the factory navigation system is available with the multimedia package. It goes for $1850. It actually comes with a Harman Kardon sound system as well but speaking of when it comes to the sound system. Eight speakers with 250 watts is the standard setup that we have today. However, like I said, there is that optional Harman Kardon sound system that comes with 13 speakers and 600 watts. That is gonna be there for you too if you wanted it. But again, we don't have that one today. So what do you guys say? Let's go ahead and turn on the radio see what we got playing today and let's test out the clarity of this one. Sound system is actually perfectly fine for the XC40. Bass is pretty good, clarity is fine. Again, this isn't that large of an SUV. So really for the XC40, that standard sound system is fine. Harman Kardon, of course it's gonna be better. Bowers and Wilkins being the very best sound system I've ever experienced. I believe it was actually a Volvo S90 that I experienced that in. That was the best I've ever heard out of the 500 reviews that I've done so far. But really with this sound system, it's perfectly fine for the XC40. But last thing I wanted to mention to you guys on the tech display at least is when you do put the XC40 in reverse, you of course will find a rear view camera with dynamic grid lines letting you know who or what is behind you, which is always is going to lead us into safety. So first thing I wanted to mention, like I was saying at the beginning of the video, the XC40 is an IIHS top safety pick, which really says it all right there. Front side, side curtain airbags come standard. In the back, you're gonna have latch, AKA lower anchors and tethers for children for the rear car seats. Real child door locks, also standard tire pressure monitoring system. Also some of the advanced safety features that come standard include lane keep assist with oncoming lane mitigation, speed limit recognition, driver attention monitor and a runoff road mitigation system. However, there is a premium package that actually adds some safety as well that goes for $1,900. That includes a blind spot monitor with rear cross traffic alert, front and rear park assist or the parking sensors essentially, and auto dimming exterior mirrors. That's something that usually comes with the auto dimming rear view mirror, but the exterior mirrors usually is an option. So that's pretty cool that that's there as well. But overall, when it comes to my final thoughts of the XC40, I love this simplistic interior design, including the 12.3 inch digital gauge cluster. Absolutely love that as well. Exterior design is very much so on point, love that. Safety is great, top safety pick of course with IHS. For me, the question comes down to, with the XC40 given its size, is it worth $40,000? And really it's a luxury compact SUV. So if you're looking for more luxury with the simplistic design digital gauge cluster, plenty of acceleration in this thing, it might be, but having said that you can get three row SUVs for around the same price. So let me know in the comment section below if you think it's worth $40,000. It really is a very nice SUV. But having said that, that is about it for this one, you guys. Thank you so much for watching. Feel free to follow me on social media at the bottom of the screen there. If you like, be sure to hit the subscribe and the bell notification button if you're into new car reviews. That is what we do here on this channel after all. Do appreciate you guys watching more than you know, and I will see you guys all in the next video. Stay Gold.


So what I want to talk about today is not necessarily language learning, but decoding what deep neural models learn about language. So as I'm sure all the audience here knows that deep neural network models are everywhere. In any domain of artificial intelligence machine learning data science when there is enough data there are enough examples of a particular task. You see traces of neural network based models that learn patterns from these large datasets and the domain of language and speech processing is no exception. Here I have put a sketch of the transformer network which is now the front runner in neural network architectures. And as you can see I'm not going to explain the architecture but what I hope that you can see from a very shallow glance is that this is a very complex architecture. On the right hand side you see a language model called birth, which is basically the foundation of many models not natural language processing models for various applications. And these birth based models have several different layers so you see that the small base birth has 12 layers and the large model has 24 layers of some part of the transformer architecture. So that's the where the power comes from, from this complexity and from the depth of these models, but with this power comes a cost. So, as I was saying, deep neural network models are very powerful but they're very hard to understand because of their complex architecture and because of the very many parameters that they have. We can't easily understand the inner dynamics of these models and this poses a challenge for both people who want to use these in applications. As well as the researchers want to understand how these models work. So some of the main questions that we are faced with are, for example, what aspects of input does a model pay attention to. What knowledge does a model learn to perform a particular task. How can we use this information to improve both the performance and the generalizability of such a model for a new domain or for the new task or new data set. So what I'm going to do today is not to provide any answers but to give you some glimpse of some general approaches that people have proposed for getting some insight into how these models work. The first approach that I want to talk about is input manipulation so this is a, an approach that's actually inspired by behavioral psychology is the idea is that you have a black box you want to understand how it operates. So you basically manipulate the input that you give to this box and then observe its behavior. For example, let's say that we want to measure how much the output of a natural language processing model changes, if you remove one word from an input sentence. Can you move to the next slide. Yeah, so as an example of a task. I think that we train a model that learns to map images to captions. So, if we give enough examples of the model it can actually map this picture of a baby laughing and looking at the computer to the caption. A baby sits on a bed laughing with the laptop computer open. Next slide. If you manage to quantify the contribution of each word, you will see that on the X axis we have each word and on the Y axis we have the contribution of that. And if you look at the blue line, which is the version that they just told you about, you see that the word baby is the most important word in this caption. So that's giving us some very general idea of what kinds of words this model's pay attention to if you are curious to see what happens if we remove the word baby from this example. Next slide will show you next slide please. Yeah, so if you remove the word baby this is the picture that the model will return so there is there seems to be a true understanding of what happens and the corresponding between words and elements in the caption. If you accumulate this kind of information. The next slide please. You will see that the model like this for example pays attention mostly to adjectives or nouns that are labeled as JJ and then and ignore certain types of words such as determiners prepositions and so on. So these kinds of input manipulations give you some insight about what types or what parts of the inputs the model actually pays more attention to and uses in the past that it performs. But it does tell us little about the actual representations that the model learns. But this takes us to the next approach when you show me the next slide. Yes, so here the idea is that we want to actually analyze these internal representations that these models learn the questions that we might have to want to ask is what aspects of language does the model encode the model encode, for example, does the model encode linguistic forms such as words morphines phonemes, syntactic structure and on which layers does this information gets encoded. Does it encode anything about the meaning and again on which layers. So one approach that one technique that has been widely used next slide please. And this is what is really called auxiliary tasks or probing classifiers. The idea here is that you have this model that you have trained to perform tasks. But now you want to know whether this model learns anything or includes anything about the knowledge why. So what you do is, you give the model original model and input. You record and extract the activations on any layer of this model, which is usually a vector of numbers and then you pass this as input to a classifier or regressor that predicts knowledge why. The argumentation here is that if you manage to train a classifier or regressor that predicts that your hypothesized knowledge with acceptable accuracy then the original model must have learned something about that type of knowledge. So here is, yeah, next slide. Here's a number of sample auxiliary tasks that you could potentially use for the example model that I showed you before mapping images through sentences. You can train classifiers or regressors to predict for example the length of the utterance that the presence of specific words representation of semantic or form similarity, etc, etc. And the next slide. This is a sample snapshot of the results of such analysis on a model that we've been working on where you see that the best performing probes probing classifiers or probing regressors for tasks that include some sort of information about linguistic form are often the ones that use input activation layers from lower layers in the model. And if you use activations or representations from top layers, then models that involve some sort of a meaning usually perform better. So these are very widespread techniques these probing classifiers or auxiliary tasks. The last approach that I wanted to tell you about is representational similarity analysis. So what happens in the case of probing classifiers or auxiliary tasks that we are trying to map representations learned by a model to a different representational space for example to the space of morphology or syntax. But when these representational spaces became become more complex or structure than these probing classifiers don't work very well. And a different type of technique that is actually borrowed from computational neuroscience comes to our rescue this is called representational similarity analysis. And the idea is that you have the same set of stimuli or data points but these are arranged differently in two different representational spaces. One is the representational space that our model learns and the other one is the target or hypothesis to representational space that we're interested in. And what we want to do is to actually measure the correlation between the similarities pairwise similarities between points across these two spaces. Next slide. So what we need here is basically a similarity metric within two spaces a and b and what we don't need is a mapping between these two spaces. So in that sense, these models are very versatile. And just as an example of their application next up next slide. So to BERT that I mentioned on the first slide to see whether it includes anything about syntactic structure and here you see that BERT the red lines. The BERT model actually includes most syntactic structure on its top layers, mainly layers 22 to 24. So to wrap up what I just said next slide. So this domain of developing interpretability techniques for deep neural models of language is a very young domain, but it's a very fastly moving and developing. And we see new techniques being introduced pretty much every day. And you can generally think of these as two different general approaches, one when we have a hypothesis in advance, we are looking for the encoding of a hypothesized type of information in our models. So probing classifiers or representation of similarity analysis are examples of this approach or data driven approaches when we look at how these representations are formed. For example by manipulating input or following the propagation in this network. I wanted to finish my presentation by telling you about this recent project that was just funded by NWA, which is exactly focused on this pursuit of interpretability techniques for deep learning models for deep models of text and sound. And last slide please. There are quite a few PIs involved in this project, but as you can see we also have three people from Tilburg, Tomlens, Czechoslovakia and myself, all from School of Humanities and Digital Sciences. So if you have any questions, I would be very happy to answer. Thank you very much.


Hello everyone, this is Amit Dhan and on this channel you'll find financial travel and more videos if you like what you see Please do the thumbs up and consider subscribing. Thank you Hello everyone, this is Amit Dhan AmitDhan.com today is my day 20 in Bangladesh and I'll show you a few clips from Dhaka, Bangladesh driving through various parts of the city You'll see the dramatic contrast between different places different streets Sometimes you'll feel like you are in a first-world country and sometimes you'll feel like you are in a third-world country, but These experiences they are totally unique which you will not find anywhere else. So if you are in Dhaka, Bangladesh Driving through the city of Dhaka, Bangladesh and checking out different parts of the city is a must and Please follow my other Another travel video and if you see what you like, please give it a thumbs up and consider subscribing because there will be more videos coming out from my Dhaka Bangladesh street. Thank you and I'll see you shortly


Dedication, Introduction, and Preface From Christopher Cricket on Cats with Observations and Deductions for the Enlightenment of the Human Race From Infancy to Maturity and even Old Age This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer, please visit LibriVox.org Christopher Cricket on Cats with Observations and Deductions for the Enlightenment of the Human Race From Infancy to Maturity and even Old Age By Anthony Henderson, you were Dedication To all the cats that ever meowed on this or any sphere From the beginning of all time, unto this present year To all the cats that still to come and to all those that lives and to their ghosts Each counten nine and to their relatives And to each one who like some sort of cats, no matter what I dedicate this little book with kind and loving thought And especially to Samuel T. Shaw Introduction Well, Pussy, as I stroke thy fur and smooth its silk and thicket I hear above thy placid purr the chirping of a cricket Ah, I suspect that subtle bug when winter skies are drabby As crawled beneath the hard stone rug to keep a tab on tabby May Pusser poet seldom thrice unkicked by rumours scandal How can a cat lead nine pure lives without breath of scandal Eight lives a cat might lead at home as blameless as St. Georgie's Yet in its night-carnation roam to witches' midnight orgies A cat of cultivated throat has repertoire's that vary Yet in the shrillness of his note he rivals the canary A cat concerto highly strung with penetrating spirit Is like some songs I too have sung, I'd rather sing than hear it So here are Master Cricket's views on cats I place reliance in what he says, though half the news he tells may startle science But should harsh critics speak of fakes, with one reply I'm ready One touch of nature faker makes the whole world kin, says Teddy Wallace Irvin Preface And bringing this work before the public It's a good thing to know that most everything here is pretty true All set some of the things Uncle Ben's helped on And if readers is ever awful doubtful in these places Guess they'd better use their own judgments Everything learned here has been the fruits of unwritten Observation, lots of asking, and a terrible lot of thinking Of course everybody knows how hard it is to find out anything And that's why this book should be of special value to students of wisdom Like Columbus, Moonshousen, Darwin, and Thompson Seaton The writer has traveled some but written lots more Loving nature like the very Dickens since his tenderest infancy Add respect and kindness for the gentle reader Scientific appellations Attenuated phraseology And very long words from foreign obsolete And barbarious languages Has been mostly expurgated Meaning to cut out, though some still remains Which if people don't understand, take the writer's fault People that's omnivorous readers Hadn't ought to read more than about just one part of the time Than think pretty hard for a long while about what they've been learning This is so they won't get mental derangement And builds up the brain cavities fine Of course everybody won't agree, maybe with all the writer's views and things Just like with other great writers That's why great writers have different views anyhow As it's not good manners to talk about yourself The reader will describe that these sentences Has been written in the third person And would have been written in the fourth If other pursuits hadn't stopped his pursuant The advanced stages of grammar It's hoped that all will get what they're looking for And lots more End of dedication, introduction, and preface Cats and Humans All the Same By Anthony Henderson Ewer Read for LibreVox.org By Anita Sloma Martinez Cats and Humans All the Same Of all the beasts that live We must, I guess, call cats the humanist For just like folks you'll find that these Has no end to their qualities For some is fat and some is lean Some very dirty and some clean Some's always licking at their fur And some's not so particular Some cats take just fresh milk and meat Some takes all that they're gave to eat Some cats they cough and some they wheeze And some's afflicted by disease Some cats is blind and stone deaf some But ain't no cat was ever dumb Some cats will monkey round and fool And play for hours with just a spool While some just sit and blink their eyes And prefer to philosophize Some cats likes most folks pretty fine But some takes care to draw the line Some comes and rubs against you which Means will you scratch them where they itch While others is so mean all through They like lots better scratching you And so you see the marvelous Wapey lines does resemble us And humans all the same just sort of Different in the name End of poem, this recording is in the public domain The Cat by Anthony Henderson Ewer Read for LibriVox.org by Mike Overby Midland Washington The cat's a four-legged quadruped Not countin' in his tail The Mrs is the Tabbycat And Thomas is the male The cat it is carnivorous although to milk and clinen It makes a hump out of its back And whiskers it looks fine in No home should be without the cat Especially where there's mouses It never goes away the cat But stays just where the house is End of poem, this recording is in the public domain Quantity of Legs by Anthony Henderson Ewer Read for LibriVox.org by Aline Quantity of Legs As stated in the foregone poem The cat's a four-legged quadruped No cat has five legs If they have, taint a quadruped cat Tripod cats is made by cutting off one leg Different breeds Cats are different breeds according to their dispositions Cats that's made for little boys and girls To mall and tees is called Maltese cats Some cats is known by their queer purrs These is called Persian cats Though most cats is kind unless molested some ain't Cats with very bad tempers is called angry cats Sometimes a very fine cat is called a magnificat In tropic lands the cat is called a purple catapult Cats with deep feelings is called feline cats Differences from other things Cats is different from horses because they have kittens Uses of cats Sometimes cats has saved the entire lives of infants By not loaning them to swallow their fur and tickling them to death Cats has been known to save hundreds of dollars worth of things From being robbed by letting burglars stumble over them in the dark And awaken up the household Principle diseases The principle diseases of cats is cataracts Catara, catagoria, catawalls, catapillars, cataba, and catalypsy Some cats is great leapers and when very joyous Jump themselves all round through the adjacent atmosphere This is called catalypsy And gets to be a disease which they throw themselves into Anytime they think of something very funny Purrins like pulse is with health And indicates the state of happiness of the purrra Just plain purrn means they've got no particular objections to anything Long as they're allowed to purr But more than 60 purrs a minute indicates their pleasure is rousing And that catalypsy is coming on if they're not sad and sufficient in time Sometimes catalypsy is just an emittment in spells But sometimes Uncle Ben says it has been known to last for days Touching the ground only at rare intervals And that often nothing was left of the victim But a few dying purrs and some fur From purrn meaning to indicate He says we derive the expression How much purr The principal product of cats is cat's eyes Cat tails Catacombs Catamarans Catagaries Catapulse Catnip And catars End of section This recording is in the public domain A poem by Anthony Henderson Heuer Read for LibriVox.org by Aline From lovely audiobooks.info Catalyptics these are Travelling to the moon See how big the trees are from the big balloon Two chaps they get scary just as they've begun Anchor drops most very quiet here in one With a sudden pull Old balloon comes taut Captain Michael Mulan says the blame things caught Someone brave and tried We'll need to see us through Captain Michael's cried up here in number two Supposing we slide down and do the best we can Take a look around and then report again Set the Freddy's while the rest all breathlessly Sat and watched them pile out here in number three Soon they found the anchor in a treetop's hide Started in to yank her all their bloom and might Bad one this is loose I go back there no more Watch me just removes He says in number four While the words he's uttering Woof, quiff, kwee Whole shebang goes sputtering Goodbye, Mr. Tree Gosh, but that was quick But now to keep alive Guess we'd better stick He says in number five Now as brave old heroes Back again they're getting Rest don't know how near those heroes came to quitten If though they'd their freedom with their funny tricks Guess you'd never see them here in number six End of poem This recording is in the public domain Special peculiarities By Anthony Henderson Ewer Read for LibriVox.org Cats talk to each other by moon From whence they get the word Communicat As cats have nine lives They have to be killed many times Before they're seriously dead Killing cats is considered unlucky Especially by cats There was a special kind of cat Having nine long tails Each tail being connected With one of its lives inside And this cat He was called a cat of nine tails These cats was used just for licking transgresses The cat being taken by their necks According to Uncle Ben And swung through the surrounding ozone Till the tails stopped at whatever portion Of the transgressa was to be licked Of course each tail that wore out Was one less life for the cat And if the neck wore out first Then the whole cat was one less So that soon there wasn't enough cats With enough tails or necks to go around All the transgresses And this special species soon became extinct Of course transgressors Didn't blame the cats so much They wouldn't have made any difference If they had I guess though If the transgressors hadn't transgressed And the cats had bit off their extra tails When they was kittens They might have all lived happy lives End of section six This recording is in the public domain Music's Debt to Cats By Anthony Henderson Ewer Read for LibriVox.org By Kim Joyce Cats is very musical They're filled with violin strings Which is coiled up like clock springs And which is worked like wind instruments By blowing through them with their livers Cats is very emotional And when agitated can run through The whole gimlet of catalined passions Both joyous, loving, and remorseful Cats is very romantic And can pose many beautiful serenades While singing without ever thinking Them up at all Many cats like to get on backyard Fences and walls And what's called nocturnes So that when something knocks them They can turn quick to the other side That's why they always sing them on walls and fences Because there's always the other side Which is very useful for turnin' purposes When there's no tellin' which side things are coming from When cats are knocked during the pursuit Of musical labours It is called a catastrophe Some people haven't any ear For cat music at all Much prefer in silence And insomnia Then there's somnambulists But they're much worse Haven't been known to eject bottles, bricks, and shoes With murderous intent Not even waitin' till the musical numbers was finished Uncle Ben says that when he was a boy Somnambulists always used bootjacks Which went straight as a gun and never missed And that the two richest men in his town Was the cat doctor and the bootjack man End of section This recording is in the public domain A Rain in Cats and Dogs By Anthony Henderson Ewer Read for LibreVox.org By Anita Sloma Martinez A Rain in Cats and Dogs Oh Uncle Ben, he says that he has sometimes seen it rain In mighty torrents from the sky With all its might and mane And God's still worse till he says That through the mists and fogs It came on down till all at once T'was rainin' cats and dogs But they was only ghosts, I guess For when the storm was done He looked around to take some home And couldn't find a one End of poem This recording is in the public domain When a cat has gone and died By Anthony Henderson Ewer Read for LibreVox.org By Trisha G When a cat has gone and died He lays down upon his side Then by ones and twos and threes Spirits ooze out by degrees Then the top one in the air Asks them all if they is there Then they answer to the roll Till just nine make up his soul Then, with a soft wailin' sound I'll start goin' round and round Chantin' while they're hoverin' Some sad and solemn requiem Then float off with direful moan Him left by his self alone End of poem This recording is in the public domain Cat's Aunt History's Page By Anthony Henderson Ewer Read for LibreVox.org By Anita Sloma Martinez Cat's Aunt History's Page The Compass Cat Cat's has wonderful bumps of directions And can tell which way the earth is No matter how far they've been fallin' from Cat's is even better than compasses Cause their hairs is all magnetic poles Which can be heard crackin' By rubbin' the fur with the human hand That's why cats can't never get lost Except when they're grounded Long, long ago Cats was used by the Chinese instead of compasses For steerin' their ships This was done by puttin' the cat On the behind end of the boat And then whichever way the cats pointed their tails They steered the ship Of course sometimes the ship's got wrecked But that wasn't the cat's fault For sometimes the wind would blow their tails Different from the way they had them pointin' Of course when cats do get lost They must always come back But if they ever don't Taint cause they can't But cause they maybe don't want to But I'd never come back if I was Some cat's that lives some places I know But cat's is wonderful good hearted that way And don't seem to mind nothin' Cat's Aunt History's Page Wild Cat's In olden times The witches was furnished with cats And at night time They used to go screeching round through the air With the witches on their backs With great big eyes Like fire in the dark and gee But they must have been terrible big cats To carry the witches Glad I wasn't there And then the witches They used to beat them with the broom switches That they had so that the cats Was just wild to get away So that's what made them wild cats Cat's Aunt History's Page The Sacred Cat Long time ago Way back in Egypt They was awful fond of cats Uncle Ben says they just worshipped them And sometimes cats was found The color of red sacs that they kept coffee in And these was called sacred cats And was specially honored by They buried alive with the dead kings and things Gee, I'd hate to have been a sacred cat Cat's Aunt History's Page The Knights of Malta In Malta Far which is the isle Out in the way that sees A breed of cats was once the style And they was called Maltese Long lean and healthful was they each And they could yell like blazes All through the night from dusk till light In ninety different places But all the people was in jobs When anger's ire was risen So once in all their nightly robes They up and got to biz Some cats was mobbed Some cats was slewn And some fled to the sea Skedaddling out into the moon Far from their loved country And folks that night they swore a oath And some crisscrossed their chests And some there was performed there both In memory of the pests To celebrate that welcome loss They called the crisscross right A Maltese cross to show how cross They was that fearful night The Knights of Malta you've all heard tell of On History's Page Well these was those that once occurred In that catastrophe age End of section This recording is in the public domain Miscellaneous observations and reasons for different things By Anthony Henderson Ewer Read for LibriVox.org by Kim Joyce Miscellaneous observations Very skinny cats is sometimes said to be cadaverous While cats can yowl for miles The way they walk so noiseless is something marvelous Trancandentalists say this is because their feet Is inhabited by the dead souls of departed beings Opposite views is held by some catropodists Who say that the reason cats walk so quiet Is just because they don't want people to know they're coming And that it's a strange fact That they really do walk as loud as cows and things That the sound is invisible to the human ear Reasons for different things Cats make most noise at night Because in the daytime there's so much else going on That people won't listen to them The reason cats can see fine at night Is their eyes is just like lamps And so they need lots of oil And that's why they eat fishes Because fishes is oily The reason cats can go anywhere at night Is because they eat lots of mouses And mouses can go anywhere at night Except where there's cats Cats don't bark Because they're afraid they might be taken for dogs Which would be very humiliating For cats are very proud of their catalyze And would rather be taken for cats Than most anything else End of section This recording is in the public domain Once there was a bulldog by Anthony Henderson Ewer Once there was a bulldog and a tabby cat Thought they'd lick each other So they had a spat When it was all over Didn't much remain Except a hunk of vertebrae And a squish of brain Like the death of Moses Nothing much is known About the main particulars Except they was alone Which was winning Victor No soul living nose But I guess to leave them Lying and repose Some day when you're passing That their spot you'll see Just some catnip growing By a dogwood tree End of poem This recording is in the public domain Catalyptic By Anthony Henderson Ewer Read for Librebox.org Once a catalytic Thinking of a joke Awful violent catalytic stroke Shout him up fruition While he flopped around Guest twist more an hour Before he hit the ground Come down all exhausted In the marsh's damp Wiggling round just like a pretzel With a cramp Gee I never again will think Of jokes like that Just one more I'm sure Would finish this here cat End of poem This recording is in the public domain Why Cats Has Whiskers By Anthony Henderson Ewer Read for Librebox.org By Sarah Brown When a cat he starts to go Through a hole somewhere He most always likes to know If there's room to spare Then he reckons how much space Does his whiskers take By just pushing through his face Measurements to make If they have to bend he knows That it means hard luck And the rest of him that goes Surely will get stuck But if they don't bend you see Tells it at a glance Knows it safe as safe can be So he takes the chance Pity people haven't all Whiskers like this chaps If they had they might forestall Many dire mishaps End of poem This recording is in the public domain Kittens By Anthony Henderson Ewer Read for Librebox.org By Elin From lovelyaudiobooks.info Kittens A funny thing about kittens Is that though they're descended from cats They begin with a K Scientists have never discovered why this is And I've been too busy to find out just why myself Kittens is just the same as pressed in cats And hens should be fed on condensed milk Kittens is stone blind for nine days After they're born Oculus and Ketnologists Disagree as to whether this is Because they wouldn't understand what things was If they did see him Asking so many questions That no mother living could answer them all At the same time Whereas Uncle Bend thinks It's just to make him glad they're living later By keeping their vision retarded Until they're able to assimilate objects and things He says that kittens always used To put him in mind of wounded soldiers The way they was both born and litters End of section This recording is in the public domain Things some people know And some things they don't By Anthony Henderson Ewer Read for LibriVox.org by Tricia G Oh yes, I almost forgot What is it that walks like a cat And has a tail like a cat And makes a noise like a cat And looks just like a cat But it ain't a cat It's a kitten Uncle Bend's great-grandfather Used to tell him one every once and a while Why is cats like poor surgeons Because they mutilate And destroy patients Then there was another that Uncle Bend Says he was brought up on Why are cats tails like eternity Because they're fur to the end Guess that's all there is To know about cats If there's any more Taint worth knowing Because I've paid particular attention to cats For almost a week And I guess I ought to know End of section This recording is in the public domain End of Christopher Cricket on Cats By Anthony Henderson Ewer


Hi, I'm Zor. Welcome to Unisor Education. I would like to dedicate this lecture to one particular property of vector products, the distributive property. Now, the distributive law of vector product in relationship to an addition of vectors has its typical form. So, if you would like to multiply to have the vector product of some two vectors by the third one, well then you multiply the first one, the second one, and then add results. Okay, that's the typical form of the distributive law. Now, I have defined the vector product in geometrical terms, namely, if you have two vectors, then to construct their vector product, you have to go to the perpendicular to both of these. So, basically, if you have a plane which is defined by these two vectors a and b, then this is a perpendicular to the whole plane. Now, the direction is defined by a smaller angle between a and b from a to b, it's important. So, you can use this the rule of the corkscrew. So, in this particular case, the direction would be upwards, and the magnitude of this vector is equal to product of the magnitude of these two vectors times the sign of the phi angle, smaller angle between a and b. Smaller, I mean, the result is an opposite angle, but I'm talking about smaller angle less than 180 degrees. So, that's the definition. Now, from this purely geometrical definition, and again, I'm emphasizing its geometrical definition, it's not coordinate form which will be addressed later. So, using this geometrical definition, it's not really obvious that the distributive law is held. So, this lecture will be about this distributive law, and I will try to prove this particular law in geometric terms, so to speak. Okay. Now, when I was trying to prove it just for myself, it turned out to be not as trivial as other properties, like anti-commutative property that AB times, BA is minus AB, something like this, or some other properties like multiplication of the vector by itself equals to zero because the sign of the angle is equal to zero. So, this is not as trivial. Now, I came up with this particular proof, and I actually put it in the notes for this lecture. I put the hint first, so kind of encouraging you to do it yourself, with or without my hint, and then I put the whole proof into the notes. So, it turned out to be not as trivial as others, but well, that's what I came up with, and if you can come up with something simpler using purely geometric approach, I would be more than happy to put it somewhere in the website or something like this. All right. So, here is my proof. First, if you don't know how to prove the general case, you probably would like to prove something in some special, a simpler case. So, what is a simpler case which I would like to consider first? Well, the simpler case is the following. So, instead of three completely different, completely general vectors a, b, and c, I will consider the following case. This is c, and this is a, which is collinear and co-directional with c, and then somewhere there is a b. Now, what I will also do, I will draw coordinates, but not because I need the coordinates, just because it would be probably easier for you to have some space feeling about the whole thing. So, b is not in the plane x, y. It's somewhere in the space, and c is along the z-axis in my particular case. I mean, I can choose whatever the coordinate system I would like to. So, I choose z as coinciding with this c vector, and in my particular case, a is collinear with c, and same direction. That's my very particular case. Now, just to show the dimensionality of this, let me just draw the projection of the b on the x, y, z. This is on the x, right? And somewhere there is a projection on the z-axis, but I don't really need it. Now, so what's interesting about this particular case? Well, the interesting thing is, obviously, that ac is zero, no vector. Why? Because they're collinear, so the sign of the angle between them is equal to zero. Doesn't matter what the lengths of them are, all right? Okay, so what follows from here is that, let's consider now what would be the a plus b. Now, a and c coincide, right? So, let's consider the plane where a and b or c and b belong to. Now, so basically what I would like to do is, I would like to put this plane on the picture, something like this. So, let's call this point b. Well, this might be a, this might be c, and this would be say d. So, I'm talking about the plane, which is defined by vectors a and b, which is the same as the plane of a rectangle O, C, B, D. So, if you can imagine, you have a vertical plane, which is turned out in some way, so it actually contains the b vector. And when this plane cuts the x or y plane, the O, D is the result of it. Try to imagine this particular thing. Now, since a and b and c all belong to the same plane, I would like to just consider what's going on in this particular plane. Let's think about it. Since a and b belong to this plane, then there's some which I can actually draw here. This would be a plus b. It also belong to the same plane. So, if I will look at this plane from a side, I will have c vector at some angle. Now, a vector is coinciding with c. At some angle, I will have b. And here I have a plus b. So, this is just the view of this plane from the side of it. Now, if I view at this plane this way, now, where are the vectors which are a, c, b, c, and a plus b, c? I'm talking about vector product. Well, a vector product c is no vector. Now, b and c, if I will multiply them, then the vector would go this way. And a plus b vector product by c also would go this way. Why? Because all of them belong to the same plane. And the vector product should be perpendicular to all the participants. So, it's perpendicular to c. And it's perpendicular to b. And it's perpendicular to a plus b. So, all these vectors, b times c, will be perpendicular to this plane. And a plus b times c will be perpendicular to this plane. But this is the same plane, so it's the same perpendicular. So what I'm talking about is that the direction of a plus b times c, or b times c, is always this way. Now, on my three-dimensional drawing, it's somewhere here. That would be dimensional. So first of all, which I would like to establish is that a times c is no vector. bc, b times c, and a plus b times c are directed in the same direction. So if I want to establish the equality between these vectors, I have to establish equality in magnitude and the direction. So direction has been already established. And it follows from the fact that c and b and a plus b all belong to the same plane. And all these results of the multiplication should be perpendicular to this plane. And there's only one perpendicular to a plane at this particular point. Now let's talk about magnitude. So what's the magnitude of, let's say, a plus b times c? Well, it would be lengths of the a plus b times lengths of the c, times sine of this angle. Let's call it alpha. And let's call this angle b. Now, the magnitude of this vector, well, a times c is no vector, so it's bc. It would be lengths of the b times lengths of the c times sine of alpha plus beta, alpha plus beta, right? From b to c is alpha plus beta. From a plus b to c is just alpha. So this is the magnitude of the left side and this is the magnitude of the right side. Are they equal? Well, let's do it this way. Let's consider this triangle. This is our point, o. This is c. And this is, I don't remember what it is, but let's call it d. Triangle, o, c, d. This is alpha. This is beta. Now, this is parallelogram because we are adding two vectors together and the rule of the parallelogram is that their sum is the diagonal. So this angle is equal to 180 minus alpha plus beta, right? 180 minus alpha minus beta. Now, the sine of pi, which is 180, minus alpha plus beta, I put alpha plus beta in parenthesis, is equal to sine of alpha plus beta, right? That's the property of the sine. And now consider in this triangle, o, c, d, the law of sines. Now, the law of sines says that every sine, every side divided by sine of the opposite angle is equal to another side to its opposite angle and the third one, which means that length of this side, which is a plus b, length of the a plus b, divided by sine of this angle, which is the same as sine of alpha plus beta, equals to this side, which is the same as this. It's length of the b, it's parallelogram, divided by sine of alpha. So that's the law of the sines. And what do we see here? This is exactly the same as the college of these guys. Well, obviously, multiplication by the length of the c doesn't change anything.  So this equality is exactly the same as equality between these things. Well, provided by the sines not equal to 0, et cetera, we are all kind of considering a general case in this case. So that actually completes the proof, because we were already proven that the vector product of b and c is this type of vector directed perpendicular to the whiteboard. And the vector product of a plus b times c, also the same thing. And their magnitude is the same. Their lengths is the same. So basically, the vectors are the same. They have the same direction and the same length. So that proves that in this particular case, let's call it case a, when the vector a is collinear and has the same direction as c. We have actually proven our theorem. It was a very easy kind of a case, a very special case to the same token, when a and c are going along the same axis. Now, let's consider case b. Now, case b would be c is perpendicular to a and perpendicular to b. So this is the case when I will also draw coordinates, although I don't really need them, just to have some kind of space representation. So these are two vectors a and b. And they belong to some plane, right? So c is perpendicular to this plane. So in theory, then imagine that you have a regular Cartesian system of co-ordinate. And within the x, y plane, you have these two vectors. They are here on this plane. And c is perpendicular along the z axis. You can consider it this way, although again, I don't really need the coordinates right now. Now, what happens in this particular case? Well, let's think about it. a times c should be perpendicular to c. All vectors perpendicular to c are in this plane, right? So a times c would probably be, and also perpendicular to a, right? So a times c would be somewhere here, let's say. This is a vector product c. Now, b vector product with c, also, the result should be perpendicular to c, which means it should be in the same plane, and also perpendicular to b. That would be my b times c. Now, a plus b, again, let's put this parallelogram. This is a plus b. a plus b is in between a and b. And actually is exactly the same thing I have to tell, that the result of the multiplication of a plus b times c should also be within this plane, because it's supposed to be perpendicular to c, and probably in between here. Now, my statement is that this parallelogram is similar to this parallelogram. Why? Well, let's think about it. This side is 90 degree from this one. This side is 90 degree from this one. And this is also 90 degree of this one. So all these angles are the same, right? Now, how about the sides? Well, angles are the same. Now, a times c, and this is a plus b times c vector product. What's their length? Well, considering that the angle between all of these and c is 90 degree, you see? The side of the angle between them is always equal to 1. So the length is actually equal to product of lengths. The length of a plus b times c is the product of lengths of a plus b and c. And same thing with this. So all these lengths are proportional to original lengths, just the scaling factor is the lengths of the c, right? Because, again, the lengths of the a product c is equal to a and c. The lengths of the b times c is b times c. Again, because the sign between them is equal to 1. It's 90 degrees. They're all perpendicular to each other. And finally, the a plus b times c would be lengths of the a plus b times lengths of the c. So all lengths are scaled by a factor of c, which means every triangle, this angle is equal to this angle. This side is proportional to this side by the factor of c. And this side is proportional to this. So all the angles, everything is exactly proportional. So this is, therefore, a diagonal. Because obviously, this side would be parallel to this one because the angles would be equal. Considering these triangles are similar, the corresponding angles are equal. And that's why it's a parallelogram. So whenever you connect points a times c, a plus b times c, and b times c, you will get the parallelogram. Because all these angles are the same and all sides are proportional. Which means that since this is the parallelogram, that actually confirms that this diagonal is actually the sum of these two vectors, which is exactly what's necessary to prove. That this one is the sum of these two vectors. So this is how we prove it purely geometrically in case when we have this type of perpendicularity. Now it's time to consider the general case. And we will use these two special cases for the proof. Now in this case, I will resort to using the coordinates. And here is how. If I have three vectors, a, b, and c, I am the one who can choose any coordinates I want in this space. So I will take the point where all of them originate. And I will put my z-axis along my c-vector. Now x and y, I will put somehow, doesn't really matter how. But obviously, x, y plane should be perpendicular to c. Now my two vectors, a and b, are somewhere in the space. Doesn't matter where. So we are talking about case c, the general case. So first of all, I choose this coordinate system where one of the z-axis actually goes along the c-vector. And a and b are somewhere in the space. Now in this coordinate system, I can represent vector a as a1, a2, a3, triplet. And vector b as b1, b2, and b3, triplet. Now let me introduce unit vectors i, j, and k along each axis. So i is a unit vector along x, j is unit vector along y, and k is unit vector along z. Now obviously, I can represent, since a1, a2, and a3 are projections. So this is a1, this is a2, and this is a3. a1, a2, and vertical a3. Now obviously, I can represent the vector a as some of these vectors. Why? For obvious reasons, vector a can be represented as some of this plus this. Now this is a3 times k. Now this can be represented as this times this. This is kj, and this is the same as this equal to, this is a2j, and this is a1i. So some of these three vectors, this one, this one, and this one gives me a. Similarly, if I will represent in the coordinate form b vector, I can represent it as a combination of coordinates multiplied by corresponding unit vectors. And finally, a plus b can be represented as some of coordinates. Let's consider what is a plus b times c. I can say that this is equal to this times c. And what I will do is the following. I will call this vector b. So I will have a1 plus b1i plus b times c. Now let's think about our first property. Now the first property says, if a is collinear with c, then this is a true statement. Now I will consider this as my a, this as my b. So a would be this one, and b would be the b. Now a and c, OK, I made a small mistake, a small mistake. Now instead of this, I forgot that I put my, I will use this as b. And this separately, because I wanted it to be collinear. All right, so it's a3 plus b3, k plus b, sorry about that. All right, so now what I'm saying is that this I will consider as my a vector for the first case. And b would be my b vector. Now you see k and c, they are collinear, because I have chosen c to be along the z-axis, right? And k is a unit vector along the same z. So basically, I can apply my formula in this particular case, because my first component in the sum is exactly collinear with c. And the second is irrelevant in this particular case. So I can't open the parentheses. I can use the distributive law, because I have proven that if the first component in the case a, if the first component is collinear with c, then I can open the parentheses. The distributive law is working, right? So I can say that a plus b product c equals to a3 plus b3k times c, not times product plus b product c. Let me open even further. Now, I know that v is a combination of i and j, which means it's completely in this horizontal plane of x and y, which means that this particular vectors, both are, so v times c is equal to a1 plus b1, a1 plus b1i plus a2 plus b2j product with c. Now, both vectors, now I consider this one as a and this one as b. Now, both vectors are perpendicular to c, because they're both in the horizontal plane. They are combinations of i and j. Therefore, I can open the parentheses here as well, because it's second theorem check problem. So that would be a1 plus b1ic plus a2 plus b2jc. So what I want to say is that I have actually opened up this in the co-ordinate form. So the whole result would be this plus a3 plus b3kc. These are just numbers. These are just numbers. And because they are numbers, I can basically separate them and multiply them. The distributive law of these numbers is working. It's with vectors I wanted to prove. But these are just numbers, which means number multiplied by something, the sum of two numbers multiplied by something, it's obviously just a vector. So it's like two different constants multiplied by some vector w. Now, with multiplication by a scalar, the distributive law works without any problems. And therefore, I can see that the whole thing, I can open all these parentheses, is equal to a1 times ic plus a plus b1. Or actually, I will group them. First, I will put all a's, and then I will put all j's, all b's, plus a3k times c. It's not really time. It's product plus b1i product c plus b2j product c plus b3k product c. So that's the result of all these multiplications. What if I will multiply separately a by c and b by c? Now, a has this form. Now, if you follow the logic, it would be exactly the same thing. First, I will separate the a3k component. So a times c would be a3k plus some vector a1i plus a2j times c. Now, I will use the first theorem, which I have proved. Since k and c are collinear, I can open using the distributive law here. It would be this plus this multiplied by c. But now we'll use the second one, because i and j are perpendicular to c. Then I can see that this is a1i c plus a2jc. So I will have this. Now, with b, I will have the corresponding thing with b. And if I will add them together, I will get exactly the same thing as on the left side here. So using these co-ordinates, which I have completely artificially created, I just chose the z axis along the c and then some perpendicular x and y. Using this and the representation of any vector in the co-ordinate form as the combination of unit vectors with the co-ordinates as multipliers. Using this, I basically have reduced my very general problem with two different cases, basically. Each one of them I have already proved. So that actually concludes this general case. And I should comment probably that this is not an easy proof, at least for me. And maybe there is something easier than that. But in any case, I would like you to concentrate on this problem, how to prove the distributive law for the vector product relative to the addition of the vectors. I would like you to concentrate on this particular topic a little bit more. Read whatever the notes are. Listen again to the lecture. It's very important. It's a great exercise for the brain for how to build this logical conclusion one after another. Because as you know, this is not some skill which you will need in your real life practice. This is just the training of your mind how to logically approach the problems like this. So I first specifically considered some very particular cases. And then the general case I have reduced to these ones. How did they come up with this? Well, difficult to say. But probably I was trying first to do some general case and then realized it's too general. And let me just simplify it. So that's probably somehow how my mind was working. But it's very individual. So don't count on some similarity or anything like this. So anyway, I would like to point you again to this lecture, to the notes for this lecture. Do it again. And you will see how logical will really fill up your mind. That's it for today. Thanks very much. And good luck.