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! =) 

See what happens when you try to take $(a\times b)\cdot a$ or $(a\times b)\cdot b$ (you should get $0$).  If a vector is perpendicular to a basis of a plane, then it is perpendicular to that entire plane.  So, the cross product of two (linearly independent) vectors, since it is orthogonal to each, is orthogonal to the plane which they span.
Also, while you're trying to develop an intuition for cross products, I highly recommend this video
https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/dot_cross_products/v/proof--relationship-between-cross-product-and-sin-of-angle
and, while we're there, might be worth knowing how the angle formula for dot products comes from the law of cosines
https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/dot_cross_products/v/defining-the-angle-between-vectors
