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.

Answer 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 $\mathbb{R}^2$.