Relationship between vectors of a triangle and dot product

Question

So I have to make this exercise:

$\vec u + \vec v + \vec w = 0$ I also know that $||\vec u|| = \frac{3}{2}, ||\vec v|| = \frac{1}{2}, ||\vec w||=2$. Now take:

$$\vec u \cdot \vec v + \vec v \cdot \vec w + \vec w \cdot \vec u $$

So, am I forgetting some identity or technique? I imagine that since $\vec u + \vec v + \vec w = 0$ they form a triangle, but I don't know how to proceed. I tough of using cosine law or something since I have the sides of the triangle, but is there a easier way to solve this exercise?

Answer 1

Hint:

$\left(\vec{u}+\vec{v}+\vec{w}\right)\cdot\left(\vec{u}+\vec{v}+\vec{w}\right)=\vec{u}\cdot\vec{u}+\vec{v}\cdot\vec{v}+\vec{w}\cdot\vec{w}+2\left[\vec{u}\cdot\vec{v}+\vec{v}\cdot\vec{w}+\vec{w}\cdot\vec{u}\right]$