Dot product cancellation

Question

Given vectors $a,b,c$, I know that $$ c\cdot a=c\cdot b $$ does not imply $a=b$ (take three orthogonal vectors, for example).

However, if I say that $c\cdot a=c\cdot b$ holds for any vector $c$, is it then true that $a=b$? How should I argue?

Answer 1

Hint: $$c\cdot a=c\cdot b\implies c\cdot a-c\cdot b=0$$ $$\implies c\cdot(a-b)=0$$

What do we infer if this equation holds for every $c$?

Answer 2

We have that

$$c\cdot a=c\cdot b \iff c\cdot (a-b)=0 \iff c=0 \quad \lor \quad a-b=0 \quad \lor \quad c\perp (a-b)$$

and since $c$ can assume any value only the case $a-b=0$ remains therefore

$$\forall c\,,\quad c\cdot a=c\cdot b \iff a=b $$

Answer 3

Hint:

If the equation holds for every $c$, you can put $c=a-b$.