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VG.4 The Cross Product

There are two ways to multiply vectors together, one is the dot product which we have already covered, and the other is cross product which we will cover now.

The Cross Product

If and , then the cross product of a and b is the vector

Notice that the result of the cross product is a vector.

The cross product is a determinant of a matrix that is composed of the two vectors and the three unit vectors:

An important property of the cross product is given below

Theorem

A way to relate the angle between two vectors and the cross product is to do the following.

Theorem

If is the angle between the vectors a and b (so 0, then

One can tell if two vectors are parallel if their cross product equals 0.

Corollary

Two nonzero vectors a and b are parallel if and only if

When looking at two vectors, one can see a parallelogram made by the two vectors with one of the vectors being the height of the parallelogram and the other vector being the base. The area of this parallelogram is determined by the magnitude of the cross product of the two vectors.

Some General properties of the cross product are:

Theorem

If a, b, and c are vectors and c is a scalar, then

Part of Equation 5 (( x ))is know as the scalar triple product. The scalar triple product is used to find the volume of a parallelepiped determined by the three vectors of the scalar triple product.

Formula

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