If you cross 2 unit vector, do you also get another unit vector at the other axis?

Question

Let's say I have 2 unit vector of X and Y. I have to find another vector of Z. So if I were to cross X and Y to get Z, will Z also be a unit vector?

Answer 1

We know that the magnitude of the cross product of $\vec x $ and $\vec y $ is given by: $$|\vec x \times \vec y| = |\vec x||\vec y| \sin \theta $$ where $\theta $ is the angle between the two vectors.

Now consider what will happen if $\vec x = \vec y $. Even if $\vec x \neq \vec y $, we do have, $|\vec x|=|\vec y| = 1$. Thus, $$ |\vec x \times \vec y| = \sin \theta $$

Hope you can take it from here.