### Projection

The*orthogonal projection*(or simply ``projection'') of onto is defined by

*coefficient of projection*. When projecting onto a

*unit length*vector , the coefficient of projection is simply the inner product of with .

**Motivation:**The basic idea of orthogonal projection of onto is to ``drop a perpendicular'' from onto to define a new vector along which we call the ``projection'' of onto . This is illustrated for in Fig.5.9 for and , in which case

**Derivation:**(1) Since any projection onto must lie along the line collinear with , write the projection as . (2) Since by definition the

*projection error*is orthogonal to , we must have

**Next Section:**

Changing Coordinates

**Previous Section:**

The Pythagorean Theorem in N-Space