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This is a Helper Page for:
Divergence Theorem
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Vectors are quantities that are specified by both a magnitude and direction. Perhaps the simplest vector is a Euclidean vector, represented by an arrow in Euclidean space. Refer to Image 1 below. This arrow has a length and points in some direction.

Image 1. Image of a vector
Image 1. Image of a vector

The length of a vector is called the magnitude. The magnitude is denoted by \left| \vec A \right|. The magnitude of a vector is a scalar quantity, a numerical value.

Image 1 is a vector. However, Image 1 is not very useful because it's unclear where it the vector is positioned. We need to introduce the Cartesian coordinate system (the x-y graph) to properly give the direction of a vector. Assigning x and y coordinates to a vector allows us to be more precise when we talk about the location of a vector. Refer to Modified Image 1 below. In this updated image, we now know the x and y coordinates. Our vector is 6 units in the x direction and 8 units in the y direction.

Image 2. Modified Image 1
Image 2. Modified Image 1

Another way to locate a vector uses unit vectors. In the two-dimensional coordinate system, the vectors \mathbf{i} = \left ( 1, 0 \right ) and \mathbf{j} = \left ( 0, 1 \right ) are our unit vectors. It's clear that unit vectors have a length of one. In textbooks, unit vectors can be written in bold text or have a hat placed above the variables like so, \hat{i}.

Any vector may be written in terms of our unit vectors \mathbf{i} and \mathbf{j} through scalar multiplication and addition (this is discussed in the graphical introduction below). For now, imagine the unit vector is a rubber substance that can stretch or shrink. With this property, we change change the magnitude of the unit vector such that it can express any vector in the coordinate system.

For example, \vec A = \left ( a_1, a_2 \right ) can be written as the following:

\left ( a_1, a_2 \right ) = \left ( a_1, 0 \right ) + \left ( 0, a_2 \right ) = a_1 \left ( 1, 0 \right ) + a_2 \left ( 0, 1 \right ) = a_1 \mathbf{i} + a_2 \mathbf{j}.

We can write the vector in the Modified Image 1 as \vec A = 6 \mathbf{i} + 8 \mathbf{j}.

Image 2: Unit Vectors (i, j, k)
Image 2: Unit Vectors (i, j, k)

In order to give a vector's position using unit vectors, we write it as a combination of unit vectors that are placed along the the coordinate axes. Unit vectors correspond to the x-y-z coordinate system this is in three-dimensions. \mathbf{i} = \left ( 1,0,0 \right ) points along the x-axis \mathbf{j} = \left ( 0,1,0 \right ) points along the y-axis and \mathbf{k} = \left ( 0,0,1 \right ) points along the z-axis. Unit vectors (sometimes called the standard basis vectors) are used in physics, engineering and linear algebra.

The \mathbf{ijk} notation is used to emphasize the "vector" nature of a vector while the coordinate notation is use to emphasize the "point" nature of a vector.

Labeling Vectors

When we have a vector quantity we put an arrow on top of the labeling letter to remind us that it is a vector. It looks like so \vec A. One way of writing vectors is by components, like this: \left ( a_1, a_2, a_3 \right ). For example, suppose we want to write a specific vector in components, and we know the vector goes 3 units in the x direction, 2 units in the y direction and 0 units in the z direction. Then we can simply write: \left ( 3, 2, 0 \right ).

The components of the same vector can also be written as: \begin{bmatrix} 3 \\ 2\\ 0\\ \end{bmatrix}. This vector has an x-component of 3, a y-component of 2 and a z-component of 0. This is shown in the image on the below.

Vector (3, 2): Click to enlarge
Vector (3, 2): Click to enlarge
With \left ( a_1, a_2, a_3 \right ) as any numerical values we can also write any vector in terms of standard unit vectors: \vec A = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}. If we are given \vec A = 3 \hat{i} + 3 \hat{j} + 2 \hat{k} we know that vector A is 3 units in the x and y direction and 2 units in the z direction. This vector could equivalently be written as \left ( 3, 3, 2 \right ) or \begin{bmatrix} 3\\ 3\\ 2\\ \end{bmatrix}. It is shown in the image below.
Vector (3,3,2): Click to enlarge
Vector (3,3,2): Click to enlarge
If a vector is still a bit abstract to you then think of a compass. The arrow has a certain length, this is our magnitude, and it points in any direction (north, south, east, and west).

Click here for a graphical introduction to vectors:

When two vectors are added, the sum can be found by placing the two vectors 'head to tail' and finding the coordinate they reach. When a vector is multiplied by a scalar several things may occur:

  • When multiplied by a number with absolute value greater than one, the vector stretches. (Figure 1)
  • When multiplied by a number with absolute value less than one, the vector shrinks. (Figure 2)
  • When multiplied by a negative number, the vector reverses direction. (Figure 3)
'Head to tail' addition of two vectors: (3, 2) + (1,-1) = (4, 1): Click to enlarge

Many mathematical and physical entities can be represented by vectors. For example, an object's velocity can be represented by a 'velocity vector', where each component represents the object's speed in a certain direction. Gravity can also be represented by a vector: the strength of gravity's pull corresponds to the magnitude of the vector, and the direction of the pull corresponds to the direction the vector points in. Vector Fields are another important application of vectors.

Figure 1: Stretching a vector A Figure 2: Shrinking a vector A Figure 3: Stretching and reversing the direction of a vector A

Click here for an algebraic introduction to vectors:

For vectors  \vec{A},\vec{B}, and  \vec{C} and numbers d and e,

The following equalities hold for all vectors and scalars:

  •  \vec{A}+\vec{B} = \vec{B} + \vec{A} (Commutivity of vector addition)
  •  \vec{A}+(\vec{B}+\vec{C}) = (\vec{A}+\vec{B})+\vec{C}  (Associativity of vector addition)
  •  d(\vec{A}+\vec{B}) = d\vec{A} + d\vec{B} (Distributivity of scalars)
  •  d(e\vec{A}) = (de)\vec{A} (Associativity of scalar multiplication)
  • the existence of a zero vector (additive identity),  \vec{0} , such that  \vec{A} + \vec{0} = \vec{A}

The standard algebraic representation of vectors is in terms of components, although not all vectors can be expressed this way:

 \vec{A} = (a_1, a_2, ... , a_k)
 \vec{B} = (b_1, b_2, ... , b_k)

Vectors in this form are added by components:

 \vec{A} + \vec{B} = (a_1, a_2, ... , a_k)+(b_1, b_2, ... , b_k)=(a_1+b_1, a_2+b_2, ... ,a_k + b_k)

Multiplying a vector by a number, known as a scalar, means multiplying each component by that number:

 d\vec{A} = (da_1, da_2, ... , da_k)

Click here for an interactive demonstration:


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