Has the operator $\frac{u \times v}{u^T v}$ a name?

Question

The operator $\frac{u \times v}{u^T v}$ gives a vector in the direction of the cross product, with magnitude equal to the tangent of the angle between $u$ and $v$. Does this thing have a name?

Answer 1

$$\vec F=\frac{\vec u \times \vec v}{\vec u. \vec v}=\tan \theta ~~\hat n$$ Here $\theta$ is the angle between $\vec u$ and $\vec v$, $\hat n$ is a unit vector along $\vec u \times \vec v.$ This vector doesn't seem to be given any name.

Answer 2

I assume it has no name.
to clarify to the commenters the norm of the vector,
its length = $\tan(the \ angle \ between \ vector \ u \ and \ vector \ v)$ = $\tan{\left(\angle(u,v)\right)}$

$\| \frac{u \ \times \ v}{u^T \ v} \| = \tan(the \ angle \ between \ vector \ u \ and \ vector \ v) = \tan{\left(\angle(u,v)\right)}$