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Generalized Theorem of Pythagoras
If u and v are orthogonal vectors in an inner product space, then
Proof The orthogonality of u and v implies that , so
Calculus Required
EXAMPLE 5 Theorem of Pythagoras in
In Example 4 we showed that and are orthogonal relative to the inner product
on . It follows from the Theorem of Pythagoras that
Thus, from the computations in Example 4, we have
We can check this result by direct integration:
Orthogonal Complements
If V is a plane through the origin of with the Euclidean inner product, then the set of all vectors that are orthogonal to every
vector in V forms the line L through the origin that is perpendicular to V (Figure 6.2.2). In the language of linear algebra we say
that the line and the plane are orthogonal complements of one another. The following definition extends this concept to general
inner product spaces.
