
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 ComponentsA 2dimensional 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 xdirection, and 4 represents the movement in the ydirection. MagnitudeTo obtain the magnitude of vector (x, y), use the distance formula: . For the vector (3, 4), this is √(3)^{2} + 4^{2} = √9 + 16 = √25 = 5. The magnitude of a vector x is written x and said, "magnitude of x". 

Article Index
Vector Manipulative
Vector Components
Vector Equality
Vector Addition
Vector Multiplication
Scalar Multiplication
Dot Product

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.

 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: 
Two vectors are equal if and only if all corresponding components are equal.
Vector 1  Vector 2  Equality 

(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 

When adding vectors, add the
corresponding
components. For example:
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. 

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 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. 
The dot product of vectors A = (a_{1}, a_{2}) and B = (b_{1}, b_{2}) is a_{1} · b_{1} + a_{2} · b_{2}. 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 ndimension vectors and is defined as:The dot product can also be calculated as
Property  Equation  Description 

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  (c_{1}a) · (c_{2}b) = (c_{1}c_{2})(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 

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. 
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