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This is a Helper Page for:
Divergence Theorem
Vector Fields
Complex Numbers
Gradients and Directional Derivatives
Change of Coordinate Systems
Math for Computer Graphics and Computer Vision
Dot Product
Image 1: Image of a vector
Modified Image 1

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

The length of a vector, which is measurable, 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 with no direction.

Image 1 is a vector. However, Image 1 is not very useful because it's unclear where it is positioned. To use vectors, we need to introduce the coordinate system. 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 on the left. 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: Unit Vectors (i, j, k)

Another way to define the position of a vector uses unit vectors. In the x-y Cartesian coordinate system, the vectors \mathbf{i} = \left ( 1, 0 \right ) and \mathbf{j} = \left ( 0, 1 \right ) are our unit vectors. In text books, unit vectors can be written in bold text or have a hat placed above the variables.

Unit vectors have a length of one. 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 so 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}.

Unit vectors correspond to the x-y-z coordinate system. \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. This notation is used in physics, engineering and linear algebra. 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.

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

Labeling Vectors

Vector (3, 2): Click to enlarge

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 \langle a_1, a_2, a_3 \right \rangle. 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 \langle 3, 2, 0 \right \rangle. 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 right.

Vector (3,3,2): Click to enlarge

With \left \langle a_1, a_2, a_3 \right \rangle 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 \langle 3, 3, 2 \right \rangle or \begin{bmatrix} 3\\ 3\\ 2\\ \end{bmatrix}. It is shown in the image below.

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
Vector bold3.PNG Vector bold4.PNG Vector bold5.PNG

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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