__Problem statement:__
The Pythagorean theorem asserts that for a set of

orthogonal vectors

,

(a) Prove this in the case

by an explicit computation of

.

(b) Show that this computation also establishes the general case by induction.

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__Attempt at solution:__
(a)

Since the dot product of two orthogonal vectors is zero, we have:

(b)

Basis step,

:

I assume that this holds up to

for

.

Now I show that it holds for

:

By orthogonality

for

:

Since the statement is true for n=1, n=k and n=k+1, it is also true for n.