Why does cross product give a vector which is perpendicular to a plane I was wondering if anyone could give me the intuition behind the cross product of two vectors $\textbf{a}$ and $\textbf{b}$. Why does their cross product $\textbf{n} = \textbf{a} \times \textbf{b}$  give me a vector which is perpendicular to a plane?
I know I can just check this by using dot product but I'm not totally satisfied with "it just works" answer =)
Thank you for any help! =) 

Look. I'm not a mathematician, but I have 
a perspective which can explain why the cross product of two vectors is another vector perpendicular to them. It is not a proof but it will help make that idea fimilar.
One can understand cross product in this way:imagine a line segment that makes colorful marks wherever it moves on a paper. Now make it move in the same direction and for the same distance as another segment, preserving the direction of the first segment. The result would be a parallelogram with area donated by ab sin(θ) where a and b are the lengths of the segments. So we can define cross product of two vectors as "the area of the parallelogram in wich these vectors are adjacent sides".
The first question is: why is it a vector? Imagine a plane containing two vectors a and b and the angle from a to b equals θ, the cross product of a and b equals ||a|| ||b|| sin(θ). That works quite well as a scalar, but the problem comes when you flip the entire plane containing the vectors, which is the same as replacing a for b and vise versa. The angle from a to b becomes -θ, and the cross product becomes ||a|| ||b|| sin(-θ), which equals -||a|| ||b|| sin(θ). Flipping the plane reverses the result. So the cross product can be represented as a vector with its starting point lying on the plane, which points "up or down". If that vector is pointing up, it will point down when you flip the plane and vise versa.
The second question is: why is the cross product perpendicular to the plane? Why doesn't it just point anywhere up or down? That's because when you flip the plane the cross product is completely reversed, which means it's perpendicular to the plane.
