Prove that if $ u \cdot v = u \cdot w $ then $v = w$ I've tried putting it up as:
$$ [u_1 v_1 + \ldots + u_n v_n] = [u_1 w_1 + \ldots + u_n w_n] $$
But this doesn't make it immediately clear...I can't simply divide by $u_1 + \ldots + u_n$ as these ($u$, $v$ and $w$) are vectors...
Any hints?

$$
u\cdot v=u\cdot w
$$
Others have shown how to show that $v=w$ if one assumes the above for all values of $u$.
To show that it's now true if one just assumes $u$, $v$, $w$ are some vectors, let's look at the circumstances in which it would fail.  Recall that $u\cdot v = \|u\| \|v\|\cos\theta$ where $\theta$ is the angle between the vectors $u$ and $v$.
Thus one circumstance in which the conclusion does not hold is when $v$ and $w$ are of equal lengths, i.e. $\|v\|=\|w\|$, and both are at the same angle with $u$.  Just draw a picture.  One can rotate $v$ about an axis in which the vector $u$ lies and get many vectors $w$ having the same length as $v$ and making the same angles with $u$.
Another circumstance in which it fails is this: picture $u$ and $v$ as arrow pointing out from the origin, and draw a plane or hyperplane at right angles to $u$ passing through the endpoint of the arrowhead of $v$.  Choose an arbitrary point in that hyperplane, and draw an arrow from the origin to that point.  Call that vector $w$.  Then show that $u\cdot v=u\cdot w$.
