What is the cross-product?

Question

What is the meaning of the cross product of two vectors? I know the dot product is what the two vectors have in common, but what physically does the cross product mean?

I understand the process and how to get a result, but what is the meaning of the result. For example, if I cross a velocity vector (East) with another velocity vector (North), I get a vector pointing out of the page. What does that vector represent, conceptually and physically?

Answer 1

The vector cross product is the product of the magnitudes of 2 vectors and the sin of the angle between them. It is orthogonal to the two original vectors and will have a sense determined by the "right-hand rule (close the fingers of the right hand from the 1st vector through the acute angle to the 2nd vector. The thumb will then point in the direction of the cross product. Note that the vector cross product is not commutative. A x B goes the opposite direction from B x A.

Answer 2

The cross product produces a 3-D vector which is orthogonal to the two vectors.

The cross product also describes a plane containing the two vectors plus the resulting vector.

http://mathworld.wolfram.com/CrossProduct.html

The cross product of east and north is "straight up" or in vector terms i x j = k where k points up from the 2D zone and establishes the 3D world.