Cross product angle formula

Question

Say there are two vectors $A$ and $B$ in $3D$.

to get the angle between the cross product of those two vectors, you use

$$||A\times B|| = ||A||\;||B||\sin(\theta). $$

right?

Is this equation equivalent to $||A\times B|| = AB$(dot product of $AB$) * sin(theta)?

Answer 1

No to both questions.

1. It's usually easier to use the dot product $A\cdot B = |A|\,|B|\,\cos\theta$.
2. It is only equivalent if either $\sin\theta=0$ or $A\cdot B=|A|\,|B|$ which happens when $\cos\theta=1$.
   (So, almost never..)