Proving Dot Product property

for vectors a, b, c:

if $a \cdot b = a \cdot c$ for all vectors a, then how would you prove $b = c$?

I thought about using a proof by contradiction, and the formula $a \cdot b = ||a|| * ||b|| cos(X)$, but I get stuck with showing $||a||cos(a,b) \neq ||b||cos(a,c)$

Apply your hypothesis to the vector $a=b-c$ in order to prove that $b-c=0(\iff b=c)$.
If you don't spot the $a=b-c$ trick, which works a treat, you can try the method you should think of if you are stuck in this kind of situation, and try choosing for $a$ the vectors you already know.
From this you conclude $b\cdot b=b\cdot c$ and $c\cdot b = c\cdot c$ so that $b\cdot b=c\cdot c$ and $b$ and $c$ have the same length.
Then use $b\cdot b=b\cdot c$ with this information to show that $\cos (b,c)=1$
This assumes $b$ and $c$ are non-zero (because we divide by the length), but if one or the other is zero, then the result follows immediately.