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Mathematical Expression Editor

We define the cross product and prove several algebraic and geometric properties.

VEC-0080: Cross Product and its Properties

In Module VEC-0050 we introduced the dot product, one of two important
products for vectors. We will now introduce the second type of product,
called the cross product. There are several important distinctions to keep in
mind. First, the dot product is defined for two vectors of , for any natural
number ; the cross-product will only be defined for vectors of . Second, the
dot product is a scalar; the cross product of two vectors will be a vector.
Finally, we will find that unlike the dot product, the cross product is not
commutative.

The cross product has many applications in physics and engineering. It also has
important geometric properties which will be addressed in this module and in
DET-0070.

Preliminaries

In order to define the cross product in a convenient way we need to define and
determinants. If you know how to find such determinants you may skip this section
and proceed directly to the definition.

Determinat A determinant is a number associated with a matrix

Determinat A determinant is a number associated with a matrix

Definition of the Cross Product

Let and be vectors in . The cross product of and , denoted by , is given by

This formula is much easier to remember when stated symbolically in terms of
determinants.

Find the cross product of and .

Properties of the Cross Product

What would happen if we took the cross product of the vectors in Example ex:crossproduct but
reversed the order?

Let and . Recall that . We need to compute . If you have already studied the effect
that switching two rows of a matrix has on its determinant, you should be able to
guess the outcome of the upcoming computation.

This computation shows that the cross product is an operation that is not
commutative. It also suggests that switching the order of the vectors changes the sign
of the result.

The next theorem lists two additional properties of the cross product. Proofs of these
properties are routine and are left to the reader. (See Practice Problems prob:scalarassocofcrossprod and prob:distofrossprod)

Let ,
and be vectors in , and be a scalar, then

(a)

Scalar Associativity

(b)

Distributivity

The cross product has several important geometric properties. The following
problems give us a glimpse of these properties.

Compute the following products:

For the two vectors in each product, sketch the vectors together with the product
vector. What do you observe about the relationship between the cross product and
the plane determined by the two vectors in the product?

. Vector is orthogonal to
both and .

Orthogonality Property

In this problem we will return to vectors of and of Example ex:crossproduct and Exploration
Problem init:crossproduct2. We know that
We will now compute the dot product of with each of the original vectors and
.
It is also easy to verify that and . Recall that the dot product of two vectors is if
and only if the two vectors are orthogonal. (Theorem th:orth) We conclude that, at least
in this case, the cross product of two vectors is orthogonal to each of the
vectors.

The following theorem establishes a relationship between the magnitude of the cross
product, the magnitudes of the two vectors involved in the cross product and the
angle between the two vectors. It is important to note that the identity in this
theorem involves the magnitude of the cross product, not the cross product
itself.

Let and be vectors in . Let be the angle between and such that . Then