In this final section of this chapter we will look at the
cross product of two vectors. We should
note that the cross product requires both of the vectors to be three
dimensional vectors.
Also, before getting into how to compute these we should
point out a major difference between dot products and cross products. The result of a dot product is a number and
the result of a cross product is a vector!
Be careful not to confuse the two.
So, let’s start with the two vectors and then the cross product is given by the
formula,
This is not an easy formula to remember. There are two ways to derive this
formula. Both of them use the fact that
the cross product is really the determinant of a 3x3 matrix. If you don’t know what this is that is don’t
worry about it. You don’t need to know
anything about matrices or determinants to use either of the methods. The notation for the determinant is as
follows,
The first row is the standard basis vectors and must appear
in the order given here. The second row
is the components of and the third row is the components of . Now, let’s take a look at the different
methods for getting the formula.
The first method uses the Method of Cofactors. If you don’t know the method of cofactors
that is fine, the result is all that we need.
Here is the formula.
This formula is not as difficult to remember as it might at
first appear to be. First, the terms
alternate in sign and notice that the 2x2 is missing the column below the
standard basis vector that multiplies it as well as the row of standard basis
vectors.
The second method is slightly easier; however, many
textbooks don’t cover this method as it will only work on 3x3
determinants. This method says to take
the determinant as listed above and then copy the first two columns onto the
end as shown below.
We now have three diagonals that move from left to right and
three diagonals that move from right to left.
We multiply along each diagonal and add those that move from left to
right and subtract those that move from right to left.
This is best seen in an example. We’ll also use this example to illustrate a
fact about cross products.
Notice that switching the order of the vectors in the cross
product simply changed all the signs in the result. Note as well that this means that the two
cross products will point in exactly opposite directions since they only differ
by a sign. We’ll formalize up this fact
shortly when we list several facts.
There is also a geometric interpretation of the cross
product. First we will let θ be the angle between the two vectors and and assume that ,
then we have the following fact,
and the following figure.
There should be a natural question at this point. How did we know that the cross product
pointed in the direction that we’ve given it here?
First, as this figure implies, the cross product is
orthogonal to both of the original vectors.
This will always be the case with one exception that we’ll get to in a
second.
Second, we knew that it pointed in the upward direction (in
this case) by the “right hand rule”.
This says that if we take our right hand, start at and rotate our fingers towards our thumb will point in the direction
of the cross product. Therefore, if we’d
sketched in above we would have gotten a vector in the
downward direction.
Example 2 A
plane is defined by any three points that are in the plane. If a plane contains the points ,
and find a vector that is orthogonal to the
plane.
Solution
The one way that we know to get an orthogonal vector is to
take a cross product. So, if we could
find two vectors that we knew were in the plane and took the cross product
of these two vectors we know that the cross product would be orthogonal to
both the vectors. However, since both
the vectors are in the plane the cross product would then also be orthogonal
to the plane.
So, we need two vectors that are in the plane. This is where the points come into the
problem. Since all three points lie in the plane any vector between them must
also be in the plane. There are many
ways to get two vectors between these points.
We will use the following two,
The cross product of these two vectors will be orthogonal
to the plane. So, let’s find the cross
product.
So, the vector will be orthogonal to the plane containing
the three points.

Now, let’s address the one time where the cross product will
not be orthogonal to the original vectors.
If the two vectors, and ,
are parallel then the angle between them is either 0 or 180 degrees. From (1)
this implies that,
From a fact about the magnitude we saw in the first section
we know that this implies
In other words, it won’t be orthogonal to the original
vectors since we have the zero vector.
This does give us another test for parallel vectors however.
Fact
Let’s also formalize up the fact about the cross product
being orthogonal to the original vectors.
Fact
Here are some nice properties about the cross product.
Properties
The determinant in the last fact is computed in the same way
that the cross product is computed. We
will see an example of this computation shortly.
There are a couple of geometric applications to the cross
product as well. Suppose we have three
vectors ,
and and we form the three dimensional figure shown
below.
The area of the parallelogram (two dimensional front of this
object) is given by,
and the volume of the parallelepiped (the whole three
dimensional object) is given by,
Note that the absolute value bars are required since the
quantity could be negative and volume isn’t negative.
We can use this volume fact to determine if three vectors
lie in the same plane or not. If three
vectors lie in the same plane then the volume of the parallelepiped will be
zero.
Example 3 Determine
if the three vectors ,
and lie in the same plane or not.
Solution
So, as we noted prior to this example all we need to do is
compute the volume of the parallelepiped formed by these three vectors. If the volume is zero they lie in the same
plane and if the volume isn’t zero they don’t lie in the same plane.
So, the volume is zero and so they lie in the same plane.
