Prove $\langle v,v\rangle=\langle w,w\rangle $ implies $v=w$

Question

This may seem rather simple, but I'm trying to rigorously confirm this intuition. Can anyone help?

Suppose v,w are elements of V. Prove: $\langle v,v\rangle=\langle w,w\rangle\ \implies v=w$.

Answer 1

Even when $V$ is one-dimensional: $$\langle -1,-1\rangle=\langle 1,1\rangle$$ but $1\ne-1$.

Answer 2

We have \begin{align} \langle v, v \rangle &= \langle w, w \rangle \iff \\ \lVert v \rVert &= \lVert w \rVert \end{align} so the vectors still can have different directions, if they have non-zero length.