If $V={v_{1},v_{2},...,v_{n}}$ is an orthogonal set, determine if $\sum_{i=1}^{n}\left\| v_{i} \right\|=\left\| \sum_{i=1}^{n}v_{i} \right\|$

Question

If $V=\{v_{1},v_{2},...,v_{n}\}$ is an orthogonal set, determine if $\sum_{i=1}^{n}\left\| v_{i} \right\|=\left\| \sum_{i=1}^{n}v_{i} \right\|$ is true or not.

I don´t even know where to start. However, before trying to solve this one, I solved the following:

Prove that: $$\sum_{i=1}^{n}\left\| v_{i} \right\|\le \left\| \sum_{i=1}^{n}v_{i} \right\|$$ I proved it using triangle inequality and induction. Are this two connected in some way? Can you solve the first one using the second one?