Orthogonal vectors and the Pythagorean theorem

__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.

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I do not have much experience with proofs, and would appreciate some help with this.

Thank you.