How do I prove that if vectors a and b are both parallel and perpendicular then at least one of them is 0?
It seems intuitive that this should be true, but I'm having difficulty finding a proof.
I know that 0 is perpendicular and parallel to every vector, and, intuitively, that it is the only such vector, but only intuitively.
Could anybody offer some help?
Best Answer
If $\vec a \parallel \vec b$, then $\vec a\cdot\vec b=\pm|\vec a||\vec b|.$
If $\vec a \perp \vec b$, then $\vec a\cdot\vec b=0.$
If both, then $\pm|\vec a||\vec b|=0$, so $|\vec a|=0$ and/or $|\vec b|=0$