Vector Multiplication


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.

    Dot Product

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:

     AB = |A| |B| cosθ

The dot product can also be calculated by

     AB = Ax Bx + Ay By + Az Bz

where vectors A and B are given as

     A = Ax i + Ay j + Az k
     B = Bx i + By j + Bz k

    Applications of the Dot Product

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|| = Au

The vector component parallel to that line is given by

     A|| = (Au) 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:


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