“Orthogonality” is immensely important. A quick scan of your current environment will undoubtedly reveal numerous surfaces and edges that are perpendicular to each other (including the edges of this page). The dot product provides a quick test for orthogonality: vectors
Given two non-parallel, nonzero vectors
Let
This definition can be a bit cumbersome to remember. After an example we will give a convenient method for computing the cross product. For now, careful examination of the products and differences given in the definition should reveal a pattern that is not too difficult to remember. (For instance, in the first component only 2 and 3 appear as subscripts; in the second component, only 1 and 3 appear as subscripts. Further study reveals the order in which they appear.)
Watch the video:
Cross Product from https://youtu.be/qsgK1d-_8ik
Let’s practice using this definition by computing a cross product.
Let
SolutionUsing Definition 11.4.1, we have
(We encourage the reader to compute this product on their own, then verify their result.)
We test whether or not
Since both dot products are zero,
We will now make a slight digression. Given four numbers
Thus
Given nine numbers
Note the minus sign in the second term. Thus
We can now express
Another way to remember the
This gives three full “upper left to lower right” diagonals, and three full “upper right to lower left” diagonals, as shown. Compute the products along each diagonal, then add the products on the right and subtract the products on the left:
This is equivalent to evaluating the determinant
We practice using this method.
Let
SolutionTo compute
We let the reader compute the products of the diagonals; we give the result:
To compute
Note how with the rows being switched, the products that once appeared on the right now appear on the left, and vice-versa. Thus the result is:
which is the opposite of
It is not coincidence that
Let
Anticommutative Property
Distributive Properties
Orthogonality Properties
Scalar Triple Product
We introduced the cross product as a way to find a vector orthogonal to two given vectors, but we did not give a proof that the construction given in Definition 11.4.1 satisfies this property. Theorem 11.4.1 asserts this property holds; we leave the verification to Exercise 47..
Property 5 from the theorem is also left to the reader to prove in Exercise 48., but it reveals something more interesting than “the cross product of a vector with itself is
(by Property 3 of Theorem 11.4.1) | ||||
(by Property 5 of Theorem 11.4.1) |
We have just shown that the cross product of parallel vectors is
Let
where
Note that this theorem makes a statement about the magnitude of the cross product. When the angle between
We demonstrate the truth of this theorem in the following example.
Let
SolutionWe use Theorem 11.3.2 to find the angle between
Our work in Example 11.4.2 showed that
Numerically, they seem equal. Using a right triangle, one can show that
which allows us to verify the theorem exactly.
The anticommutative property of the cross product demonstrates that
Another property of the cross product, as defined, is that it follows the right hand rule. Given
For vectors
There are a number of ways in which the cross product is useful in mathematics, physics and other areas of science beyond “just” finding a vector perpendicular to two others. We highlight a few here.
It is a standard geometry fact that the area of a parallelogram is
(11.4.1) |
where the second equality comes from Theorem 11.4.2. We illustrate using Equation (11.4.1) in the following example.
(a)
Find the area of the parallelogram defined by the vectors
(fullscreen)
(b)
Figure 11.4.3: Sketching the parallelograms in Example 11.4.4.¶
Solution
(a)
Figure 11.4.3(a) sketches the parallelogram defined by the vectors
This application is more commonly used to find the area of a triangle (because triangles are used more often than parallelograms). We illustrate this in the following example.
Find the area of the triangle with vertices
SolutionWe found the area of this triangle in Example 6.1.5 to be
We can choose any two sides of the triangle to use to form vectors; we choose
We arrive at the same answer as before with less work.
The three dimensional analogue to the parallelogram is the parallelepiped.
Each face is parallel to the opposite face, as illustrated in Figure 11.4.5. By crossing
Thus the volume of a parallelepiped defined by vectors
(11.4.2) |
Note how this is the Scalar Triple Product, first seen in Theorem 11.4.1. Applying the identities given in the theorem shows that we can apply the Scalar Triple Product in any “order” we choose to find the volume. That is,
As with the cross product, we can also write
Because
††margin:
(fullscreen)
Figure 11.4.6: A parallelepiped in Example 11.4.6.¶
the volume is the absolute value of the determinant, changing the order of the rows can only change the sign of this determinant, which doesn’t change the final answer.
Find the volume of the parallelepiped defined by the vectors
SolutionWe apply Equation (11.4.2). We first find
So the volume of the parallelepiped is 2 cubic units. In terms of determinants, we have
and the absolute value of this determinant is again 2.
Torque is a measure of the turning force applied to an object. A classic scenario involving torque is the application of a wrench to a bolt. When a force is applied to the wrench, the bolt turns. When we represent the force and wrench with vectors
While a full understanding of torque is beyond the purposes of this book, when a force
(11.4.3) |
A lever of length 2ft makes an angle with the horizontal of
the force is perpendicular to the lever, and
the force makes an angle of
Solution
We start by determining vectors for the force and lever arm. Since the lever arm makes a
This clearly has a magnitude of 20 ft-lb.
We can view the force and lever arm vectors as lying “on the page”; our computation of
Our lever arm can still be represented by
We again make the third component 0 and take the cross product to find the torque:
As one might expect, when the force and lever arm vectors are orthogonal, the magnitude of force is greater than when the vectors are not orthogonal.
While the cross product has a variety of applications (as noted in this chapter), its fundamental use is finding a vector perpendicular to two others. Knowing a vector is orthogonal to two others is of incredible importance, as it allows us to find the equations of lines and planes in a variety of contexts. The importance of the cross product, in some sense, relies on the importance of lines and planes, which see widespread use throughout engineering, physics and mathematics. We study lines and planes in the next two sections.
The cross product of two vectors is a , not a scalar.
Give a synonym for “orthogonal.”
T/F: A fundamental principle of the cross product is that
is a measure of the turning force applied to an object.
T/F: If
In Exercises 9–12., calculate the determinant.
In Exercises 13–20., vectors
Pick any vectors
Pick any vectors
In Exercises 23–26., the magnitudes of vectors
In Exercises 27–30., find the area of the parallelogram defined by the given vectors.
In Exercises 31–34., find the area of the triangle with the given vertices.
Vertices:
Vertices:
Vertices:
Vertices:
In Exercises 35–36., find the area of the quadrilateral with the given vertices. (Hint: break the quadrilateral into 2 triangles.)
Vertices:
Vertices:
In Exercises 37–38., find the volume of the parallelepiped defined by the given vectors.
In Exercises 39–42., find a unit vector orthogonal to both
A bicycle rider applies 150lb of force, straight down, onto a pedal that extends 7in horizontally from the crankshaft. Find the magnitude of the torque applied to the crankshaft.
A bicycle rider applies 150lb of force, straight down, onto a pedal that extends 7in from the crankshaft, making a
To turn a stubborn bolt, 80lb of force is applied to a 10in wrench. What is the maximum amount of torque that can be applied to the bolt?
To turn a stubborn bolt, 80lb of force is applied to a 10in wrench in a confined space, where the direction of applied force makes a
Show, using the definition of the Cross Product, that
Show, using the definition of the Cross Product, that
Suppose
Show that
Show that
Show that
Show that
Show that for any real numbers
Show that
Show that
Show that
We have seen that if we swap two rows of a
We have seen that if two rows of a