Proof of dot product = 0 (orthogonality?)

Question

Let $x_1,x_2$ be in $\mathbb{R}^n$ How can I prove that if $$\|x_1 + x_2\|^2 = \|x_1\|^2 + \|x_2\|^2$$

then the dot product of the vectors; $x_1\cdot x_2 = 0$.

Answer 1

Hint: $\lVert \vec v \rVert^2 = \vec v \cdot \vec v$ for all $\vec v \in \mathbb{R}^n$; apply this to the three vectors $\vec x_1$, $\vec x_2$ and $\vec x_1+\vec x_2$.