### The Pythagorean Theorem in N-Space

In 2D, the Pythagorean Theorem says that when and are orthogonal, as in Fig.5.8, (*i.e.*, when the vectors and intersect at a

*right angle*), then we have

This relationship generalizes to dimensions, as we can easily show:

If , then and Eq.(5.1) holds in dimensions. Note that the converse is not true in . That is, does not imply in . For a counterexample, consider , , in which case

**Next Section:**

Projection

**Previous Section:**

Orthogonality