# Vector Projection

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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.

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Note that this final formula for puv already assumes that v is normalized. If v is not normalized, then you have to divide by the square of |v|, instead of just |v|. So, if v is not normalized, you need to use the formula:

puv = ( Dot(v,u) / (|v|^2) ) * v

send me worked examples