There is a second way to multiply two vectors. This product of two vectors produces a third vector, which is why it is often referred to as ``the'' vector product (even though there are a number of products involving vectors). It is symbolically differentiated by the multiplication symbol used, which is a large sign, hence it is often referred to as the cross product both for the (cross-like) shape of this sign and because of the pattern of multiplication of components. We write the cross product of two vectors as e.g. .
The cross product anticommutes:
It is distributive:
(although the order of the product must be maintained!)
It as noted above produces a vector (really a pseudovector, explained later) from two vectors. The magnitude of the cross product of two vectors is defined by:
using terms similar to those used above in our discussion of dot products.
The direction of is given by the right-hand rule. The direction is always perpendicular or normal to the plane defined by the two non-colinear vectors in the cross product. That leaves two possibilities. If you let the fingers of your right hand line up with (the first) so that they can curl through the small angle (the one less than that will not hurt your wrist) into then the thumb of your right hand will pick out the perpendicular direction of the cross product. In the figure above, it is out of the page.
Together with the rule for rescaling vectors this proves that the cross product of any vector with itself or any vector parallel or antiparallel to itself is zero. This also follows from the expression for the magnitude with or .
Let us form the Cartesian representation of a cross product of two vectors. We begin by noting that a right handed coordinate system is defined by the requirement that the unit vectors satisfy:
This is illustrated here:
You can easily check that it is also true that:
We use the anticommution rule on these three equations:
And note that:
This forms the full multiplication table of the orthonormal unit vectors of a standard right-handed Cartesian coordinate system, and the Cartesian (and various other orthonormal) coordinate cross product now follows.
Applying the distributive rule and the scalar multiplication rule,
multiply out all of the terms in
The diagnonal terms vanish. The other terms can all be simplified with the unit vector rules above. The result is:
This form is easy to remember if you note that each leading term is a cyclic permutation of xyz. That is, , and are yzx, zxy, and xyz. The second term in each parentheses is the same as the first but in the opposite order, with the attendant minus sign, from the cyclic permutations of zyx.