# Engineering Analysis/Projections

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## Projection[edit | edit source]

The projection of a vector onto the vector space is the minimum distance between v and the space *W*. In other words, we need to minimize the distance between vector v, and an arbitrary vector :

[Projection onto space W]

For every vector there exists a vector called the projection of v onto W such that <v-w, p> = 0, where p is an arbitrary element of *W*.

### Orthogonal Complement[edit | edit source]

## Distance between v and W[edit | edit source]

The distance between and the space *W* is given as the minimum distance between v and an arbitrary :

## Intersections[edit | edit source]

Given two vector spaces *V* and *W*, what is the overlapping area between the two? We define an arbitrary vector z that is a component of both *V*, and *W*:

Where N is the nullspace.