If the scalar product of two vectors is equal to the magnitude of their vector product, find the angle between them.

Question

I found some similar questions but none could satisfy me. I am not given any other conditions except those mentioned above. Please help me with this question. I am mainly confused on the formula for the magnitude of vector product between 2 vectors.

Answer 1

With two vectors $u,v\in\Bbb R^3$ and angle $\alpha$ between them, We know $$ u\cdot v = |u||v|\cos\alpha$$ and $$ |u\times v|=|u||v|\sin \alpha$$ Hence the given conditions imply $$ |u||v|\cos\alpha=|u||v|\sin \alpha.$$ This holds trivially when at least one of $u,v$ is the zero vector (in which case we say they are orthogonal, though actually speaking of an angle between them makes no sense). If we only consider the case that $u,v$ are both non-zero, we can divide by the lengths and find $$ \cos\alpha=\sin\alpha.$$ As additionally $0\le\alpha\le\pi$, the only solution to this is $$\alpha=\frac\pi4,$$ where $\sin\alpha=\cos\alpha =\frac{\sqrt 2}2$.