

In the previous two sections, addition and subtraction of 2D and 3D vectors were illustrated. When "multiplying" two vectors, a special types of multiplication must be used, called the "Dot Product" and the "Cross Product". This section deals with only the dot product. The cross product is presented in a later section. 
Dot Product Angle 

The dot product of two vectors A and B is defined as the product of the magnitudes A and B and the cosine of the angle θ
between them:
A • B = A B cosθ
The dot product can also be calculated by
A • B = A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z}
where vectors A and B are given as
A = A_{x} i + A_{y} j + A_{z} k
B = B_{x} i + B_{y} j + B_{z} k

Component Parallel to a Line
Component Perpendicular to a Line 

Using the dot product, the angle between two known vectors
A and B, can be determined as
If the direction of a line is defined by the unit vector u, then the scalar component of the vector A parallel to that line is given by
A_{} = A • u
The vector component parallel to that line is given by
A_{} = (A • u) u
Using the properties of vector addition, or the Pythagorean theorem, we can also determine the scalar and vector components of A perpendicular to the line:
