I'm reading about angular momentum in physics, and we are given the definition that
$\vec{\tau} = \vec{r} \times \vec{F}$
where $\vec{\tau}$ is torque, $\vec{r}$ is displacement from the origin, and $\vec{F}$ is an incident force.
They later use this result in problem solving, but rather than using a cross product, they simply say
$\tau = Fr$
where these are no longer vectors.
My question is simply this; if all we care about is the magnitude of the result of a cross product, can we just take the answer as a normal scalar product? That is, is it true that
$|\vec{a} \times \vec{b}| = |\vec{a}|*|\vec{b}|$
which is what they seem to have done here?
No. The magnitude of the cross product is
$ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin(\theta)$
where $\theta$ is the angle between the vectors. So $\tau = Fr$ is true for $\theta = \pi / 2$.
However, the magnitude of the scalar product is different, it is $|\vec{a} \cdot \vec{b}| = |\vec{a}| |\vec{b}| \cos(\theta)$.