vector times cross product 
We have vectors $x, y, z$ where $z = x \times y$.

What is $x \cdot z$?
From my intuition, the cross product is perpendicular to both vectors, so dot product should be 0?

Correct. If $x$ is not collinear to $y$,  $z$ defines the direction of a normal vector to the plane that both $x$ and $y$ lie in. So $z$ is normal to both. In the case where $x$ and $y$ are collinear (one is a scalar multiple of the other),  then the cross product between them is the null vector, so the assertion is trivially true.
