Vector Projection

Consider two vectors v and u. The purpose of this section is to show how to compute the projection of vector u onto vector v.

The vector puv is the projection of vector u on vector v.

As v and puv share the same direction, and assuming the v is normalized, puv can be defined as:

where |puv| stands for the length of puv. So finding out |puv| allows us to easily find vector puv. The relation between the length of u and puv is given by the cosine of the angle between them.

The definition of the dot product says that

Hence, the value of the length of vector puv is:

So, looking back at the first equation, vector puv is defined as:

If vector v is normalized, i.e. it has unit length, then the division can be spared.