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
while

.
For real vectors

, the Pythagorean theorem Eq.

(
5.1)
holds if and only if the vectors are orthogonal. To see this, note
that, from Eq.

(
5.2), when the Pythagorean theorem holds, either

or

is zero, or

is zero or purely imaginary,
by property 1 of
norms (see §
5.8.2). If the
inner product
cannot be imaginary, it must be zero.
Note that we also have an alternate version of the Pythagorean
theorem:
Next Section: ProjectionPrevious Section: Orthogonality