Vector

Pronunciation: /ˈvɛk.tər/ Explain
A vector AB with a tail A and a head B is drawn as an arrow from point A to point B.
Figure 1: Vector AB

Multiple representations of the same vector. This vector can be drawn anyplace on the graph.
Figure 2: Vector without location

A vector is a value that has both magnitude and direction. See figure 1. The magnitude is represented by the length of the line. The direction is represented by by the rotation, or angle, of the line from a reference direction. By convention, the reference direction is usually the horizontal axis from the origin to the right.

A vector is drawn as an arrow going from the tail of the vector to the head of the vector. The tail is where the vector starts, and the head is where the vector ends. See figure 1.

A vector does not have a location. This means that the same vector can be drawn anywhere on a graph. Each of the representations of a vector in figure 2 are the same vector. Each of these representations has the same magnitude and direction so they are the same vector.

Vector Components

A 2-dimensional vector can also be expressed as an ordered pair such as (-3, 4). Each of the numbers is called a component. For the vector (-3, 4), the -3 represents the movement in the x-direction, and 4 represents the movement in the y-direction.

Magnitude

To obtain the magnitude of vector (x, y), use the distance formula: sqrt(x^2+y^2). For the vector (-3, 4), this is √(-3)2 + 42 = √9 + 16 = √25 = 5. The magnitude of a vector x is written |x| and said, "magnitude of x".
The vector <4,-3> has a magnitude of 5
Figure 3: Graph of the vector >4,-3<

Manipulative

Click on the blue points and drag them to change the figure.

What can you say about points A and B if the magnitude of vector v is zero?
Manipulative 1 - Vector Created with GeoGebra.

In the manipulative, the red arrow is the vector. Point A is the starting point, or tail, of the vector. Point B is the ending point, or head, of the vector. The green dashed vectors are the horizontal component and vertical component of vector AB.

Here are some suggestions for exploring vectors.

  • Move point A to (1,1). Move point B to (2,2). Look at the value of the vector. It should be (1,1). Now move point A to (1,2). Move point B to (2,3). Look at the value of the vector. It is still (1,1). Why did the vector value not change?
  • Now move point B to (0,1). Look at the value of the vector. It should be (-1,-1). If two vectors of the same length are pointing in opposite directions, what will be their values? If one of the vectors is (a, ), what will be the value of the other vector?

Direction

Magnitude and direction of vector (2,3).
Figure 4: Magnitude and direction

The direction of a vector can be expressed as an angle from the horizontal axis. Given the components of a vector, the direction can be calculated using the definition of tangent. Using figure 4:
tan(?)=y/x implies tan(?)=3/2 implies ?=arctan(3/2) implies ? is about .983 rad or 56.3°

Vector Equality

Two vectors are equal if and only if all corresponding components are equal.

Vector 1Vector 2Equality
(3, 4)(3, 4) The vectors are equal because 3 = 3 and 4 = 4.
(-3, 3)(3, 2) The vectors are not equal because -3 ≠ 3 and 3 ≠ 2.
(-2, 1)(-2, 2) The vectors are not equal. Even though -2 = -2, 1 ≠ 2.
(3, -1)(2, -1) The vectors are not equal. Even though -1 = -1, 3 ≠ 2.
Table 1: Vector Equality

Vector Addition

Two vectors (2,-1) and (3,1) being placed head to tail to show addition.
Figure 4: Vector Addition

When adding vectors, add the corresponding components. For example:
(2, -1) + (3, 1) = (2 + 3, -1 + 1) = (5, 0)

When representing vector addition graphically, draw the tail of one vector at the same place as the head of the other (see figure 4). When we add numbers, it doesn't matter which number we add first, 1 + 2 = 2 + 1 = 3. It is the same with vectors. Figure 5 shows the addition of vectors (2, 1) and (1, 3). No matter which one we apply first, the result is always (3, 4).

Vector subtraction is done the same way as addition. However, when subtracting vectors, it does matter which comes first.

Graphical representation of (2,1)+(1,3) and (1,3)+(2,1) showing the result is the same for both addition problems.
Figure 5: Vector Addition

Vector Multiplication

There are three types of vector multiplication that are defined: scalar multiplication, the dot product and the cross product. The cross product is out of the scope of this encyclopedia. See More Information for more information on the cross product.

Scalar Multiplication

Vector (-1,2) and (-2,4) showing that 2*(-1,2)=(-2,4) has the same direction, but is twice as long.
Figure 6: Scalar Multiplication by 2.

Scalar multiplication of vectors consists of multiplying a vector by a number. For example, the statement 3 · (-1, -2) is multiplying the vector (-1, -2) by the number 3. In scalar multiplication, each component of the vector is multiplied by the number. So, 3 · (-1, -2) = (3 · -1, 3 · -2) = (-3, -6).

Notice in figure 6, that scalar multiplication does not change the direction of the vector. It does change the length of the vector. When multiplying a vector by 2, the resulting vector will be twice as long.

Dot Product

The dot product of vectors A = (a1, a2) and B = (b1, b2) is a1 · b1 + a2 · b2. For the vectors A = (-1, 2) and B = (2, 3), the dot product is (-1) · 2 + 2 · 3 = -2 + 6 = 4. The dot product can also be called the scalar product. Note that a scalar product is not the same thing as scalar multiplication.

The formal mathematical definition of a dot product between two n-dimension vectors a = [a1, a2, a3, ..., an] and b = [b1, b2, b3, ..., bn] is defined as:
a dot b = sum for i = 1 to n of ai*bi = a1*b1 + a2*b2 + a3*b3 + ... + an*bn.
Stated in plain language, a dot product of two vectors is the sum of the products of corresponding elements of the vectors.

The dot product can also be calculated as a dot product b = magnitude of a times the magnitude of b times cos of the angle between the vectors.

Properties of the Dot Product

PropertyEquationDescription
Commutative a · b = b · a The vector dot product is commutative.
Distributive a · (b + c) = a · b + a · c The vector dot product is distributive over vector addition.
Scalar Multiplication (c1a) · (c2b) = (c1c2)(a · b) A scalar multiplied by a dot product can be pulled out so the dot product can be performed.
Perpendicular Vectors a is perpendicular to b if and only if a · b = 0 Two vectors are perpendicular if and only if their dot product is zero.
Cancellation law does not apply b = c does not imply a · b = a · c. In multiplication of real numbers ab = ac if and only if b = c. This does not apply to the dot product of vectors.
Table 2: Properties of the Dot Product

Collinear Vectors

Click on the blue points and drag them to change the figure.

Can you move the points so that the vectors go in the opposite direction.
Manipulative 2 - Collinear Vectors Created with GeoGebra.

Two or more vectors are collinear if they go in the same direction or in opposite directions. Vectors that are nonzero scalar multiples of each other. Definition: Vector u is collinear with vector v if and only if, for some nonzero real number a, u = a · v.

References

  1. McAdams, David E.. All Math Words Dictionary, vector. 2nd Classroom edition 20150108-4799968. pg 189. Life is a Story Problem LLC. January 8, 2015. Buy the book

Cite this article as:

 McAdams, David E. Vector. 5/13/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/v/vector.html.

Image Credits

Revision History

5/13/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
12/17/2018: Removed broken links, updated license, implemented new markup, update to new GeoGebra app. (McAdams, David E.)
8/28/2018: Corrected spelling. (McAdams, David E.)
8/7/2018: Changed vocabulary links to WORDLINK format. (McAdams, David E.)
12/1/2009: Added dot product and revisions for clarification. (McAdams, David E.)
9/16/2008: Added manipulative. (McAdams, David E.)
4/4/2008: Expanded article. (McAdams, David E.)
3/11/2008: Added reference to location and vocabulary hot links. (McAdams, David E.)
7/12/2007: Initial version. (McAdams, David E.)

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