Sin in Cross Product Formula

Question

I know that $ |A\times B| = ||A|| ||B|| \sin(\theta)$, wherein the case of 3D Vectors and I wanted to calculate $\sin(\theta)$ from the previous formula I will get a [1x3] vector as a result, my question is, can the 3 values in the resulted sin vector be treated as 3 separate angles with the primary axes, say $X, Y, Z$?

Answer 1

Your formula is not correct. It should be$$\|A\times B\|=\|A\|\|B\|\sin(\theta)$$and therefore, unless $A=(0,0,0)$ or $B=(0,0,0)$, you can compute $\sin\theta$ by doing$$\sin(\theta)=\frac{\|A\times B\|}{\|A\|\|B\|}.$$