Is the result of **u** **×** **u** = **0**
perpendicular to
both operands?

Yes, because **0 · u** = 0, which means that
they are perpendicular.

(Recall that by definition, the vector **0** is perpendicular to every vector.)

What about
**u** **×** **0** ?
The magnitude
|**u**||**0**| sin θ
is zero,
and the result is a vector, so:

u×0=0×u=0

Now, consider
(k**u**) **×** **v**.
Plug into the formula
for the magnitude,
|**u**||**v**| sin θ,
see what you get.

What is (k**u**) **×** **v** ?