In the previous two sections, addition and subtraction of 2-D and 3-D 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.

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 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: