Inner product space

[aristarty]

Hi everyone!
I faced with a problem: prove that two vectors of inner product space is on the same ray only when $\left \| x+y \right \| = \left \| x \right \| + \left \| y \right \|$.
Does anyone know how to prove it?

[Tolaso J Kos]

Hint: Equality holds when vectors are parallel i.e, $u=kv$, $k \in \mathbb{R}^+$ because $u \cdot v= \|u \| \cdot \|v\| \cos \theta$ when $\cos \theta=1$, the equality of the Cauchy-Schwarz inequality holds.