How do I prove that the vectors $w,\,u,\,v$ are positively oriented, if $w$ is the cross product of $u$ and $v$?

Question

Pretty much the question in the title. My textbook claims (without a reference or even an argument) that

1. The cross product $w$ of vectors $u$ and $v$ is orthogonal towards both vectors.
2. $||w||=||u||||v||\sin \alpha$
3. If vectors $u$ and $v$ are not parallel, the vector triple $u,\,v,\,w$ is positively oriented.

I managed to prove the first claim myself, and the second through this wonderful video by khan academy. I have no idea however of how I'm going to prove the third claim, and my web searches have come up short.

Any ideas?

Answer 1

Hint: work in a coordinate system where$$u=\Vert u\Vert\vec{i},\,v=a\vec{i}+b\vec{j},\,a^2+b^2=\Vert v\Vert^2$$so $w=\Vert u\Vert b\vec{k}$. Now separately consider the cases $b<0,\,b=0,\,b>0$.