Deduce the Pythagorean Theorem

Question

Let V be an inner product space,and suppose that $x$ and $y $ are orthogonal vectors in V.We also know $\left\lVert x+y \right\rVert^2=\left\lVert x \right\rVert^2+\left\lVert y \right\rVert^2$.My question is how can we deduce the pythagorean theorem in $\mathbb R^2$ from it.If possible give geometrical views.

Any help will be greatly appreciated.

thanks!! in advance.

R^2 is an inner product space - done. – Chill2Macht May 03 '16 at 21:10 — Chill2Macht, May 03 '16 at 21:10

nivekgnay · Answer 1 · 2016-05-03T21:20:18.577

1

Use the standard inner product space with $\mathbb{R^2}$. Then $\left\lVert x \right\rVert$ is the length of x, $\left\lVert y \right\rVert$ is the length of y, and $\left\lVert x+y \right\rVert$ is the length of the diagonal connecting $x$ and $y$. In this case then, $\left\lVert x+y \right\rVert^2=\left\lVert x \right\rVert^2+\left\lVert y \right\rVert^2$ is equivalent to the pythagorean theorem in $\mathbb{R}^2$.

edited May 03 '16 at 21:20

answered May 03 '16 at 21:07

nivekgnay

653

:Thanks!!.Actually,i'd taken the length of x to be IIxII instead of IIxII^2 that's why i get confused. – Styles May 03 '16 at 21:11
@PKStyles no problem, good luck! – nivekgnay May 03 '16 at 21:14
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There does seem to be some confusion here. $||x||$ should indeed be the length of $x$. – Lee Mosher May 03 '16 at 21:15
@LeeMosher you are right, thank you. I have edited my answer. – nivekgnay May 03 '16 at 21:20
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@PKStyles sorry for the confusion, ||x|| is indeed the length of x. – nivekgnay May 03 '16 at 21:21

Deduce the Pythagorean Theorem

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