Dot product cancellation 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?

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