How to prove Vector Angles? 
Any vector $\mathbf{v}$ is a unit vector if $\|\mathbf{v}\| = 1$.
Let $\mathbf{x}$, $\mathbf{y}$, and $\mathbf{z}$ be unit vectors, such
  that $\mathbf{x} + \mathbf{y} + \mathbf{z} = \mathbf{0}$. Show that
  the angle between any two of these vectors is $120^\circ$.

I know how to prove this using geometry, but the problem instructs me not to. How should I start to/prove this?
Thanks in advance!

We have that $$(\mathbf{x} + \mathbf {y} + \mathbf {z})^2 =0$$ $$\Rightarrow \mathbf {x}^2 + \mathbf {y}^2 + \mathbf {z}^2 +2 (\mathbf {x}\cdot \mathbf {y} + \mathbf {y}\cdot \mathbf {z} + \mathbf {z}\cdot \mathbf {x}) =0$$ $$\Rightarrow (\mathbf {x}\cdot \mathbf {y} + \mathbf {y}\cdot \mathbf {z} + \mathbf {z}\cdot \mathbf {x}) = -\frac {3}{2} $$ $$(|\mathbf {x}||\mathbf {y}|\cos \theta_{xy} + |\mathbf {y}||\mathbf {z}|\cos \theta_{yz} + |\mathbf {z}||\mathbf {x}|\cos \theta_{zx}) = -\frac {3}{2} $$ $$=\cos \theta_{xy} + \cos \theta_{yz} + \cos \theta_{zx} = -\frac {3}{2}$$
Can you conclude using this? Hope it helps. 
