Section 1.4 The Cross Product
¶Motivating Questions
How and when is the cross product of two vectors defined?
What geometric information does the cross product provide?
The last two sections have introduced some basic algebraic operations on vectors—addition, scalar multiplication, and the dot product—with useful geometric interpretations. In this section, we will meet a final algebraic operation, the cross product, which again conveys important geometric information.
To begin, we must emphasize that the cross product is only defined for vectors \(\vu\) and \(\vv\) in \(\R^3\text{.}\) Also, remember that we use a righthand coordinate system, as described in Section 1.1. In particular, recall that the vectors \(\vi\text{,}\) \(\vj\text{,}\) and \(\vk\) are oriented as shown below in Figure 1.4.1. Earlier, we noticed that if we point the index finger of our right hand in the direction of \(\vi\) and our middle finger in the direction of \(\vj\text{,}\) then our thumb points in the direction of \(\vk\text{.}\)
Preview Activity 1.4.1.
The cross product of two vectors, \(\vu\) and \(\vv\text{,}\) will itself be a vector denoted \(\vu\times\vv\text{.}\) The direction of \(\vu\times\vv\) is determined by the righthand rule: if we point the index finger of our right hand in the direction of \(\vu\) and our middle finger in the direction of \(\vv\text{,}\) then our thumb points in the direction of \(\vu\times\vv\text{.}\)
We begin by defining the cross products using the vectors \(\vi\text{,}\) \(\vj\text{,}\) and \(\vk\text{.}\) Referring to Figure 1.4.1, explain why \(\vi\text{,}\) \(\vj\text{,}\) \(\vk\) in that order form a righthand system. We then define \(\vi \times \vj\) to be \(\vk\) — that is \(\vi\times\vj = \vk\text{.}\)
Now explain why \(\vi\text{,}\) \(\vk\text{,}\) and \(\vj\) in that order form a righthand system. We then define \(\vi \times \vk\) to be \(\vj\) — that is \(\vi\times\vk=\vj\text{.}\)
Continuing in this way, complete the missing entries in Table 1.4.2.
\(\vi\times\vj = \vk\) \(\vi\times\vk = \vj\) \(\vj\times\vk =\) \(\vj\times\vi =\) \(\vk\times\vi =\) \(\vk\times\vj =\) Table 1.4.2. Table of cross products involving \(\vi\text{,}\) \(\vj\text{,}\) and \(\vk\text{.}\) Up to this point, the products you have seen, such as the product of real numbers and the dot product of vectors, have been commutative, meaning that the product does not depend on the order of the terms. For instance, \(2\cdot5 = 5\cdot 2\text{.}\) The table above suggests, however, that the cross product is anticommutative: for any vectors \(\vu\) and \(\vv\) in \(\R^3\text{,}\) \(\vu\times\vv = \vv\times\vu\text{.}\) If we consider the case when \(\vu=\vv\text{,}\) this shows that \(\vv\times\vv = (\vv\times\vv)\text{.}\) What does this tell us about \(\vv\times\vv\text{;}\) in particular, what vector is unchanged by scalar multiplication by \(1\text{?}\)

It is not difficult to show that the cross product interacts with scalar multiplication and vector addition as one would expect: that is
\begin{align*} (c\vu) \times \vv =\mathstrut \amp c(\vu \times \vv) \\ (\vu + \vv) \times\vw =\mathstrut \amp (\vu\times\vw) + (\vv\times\vw) \end{align*}We can combine these properties to make cross product calculations a bit easier. For example,
\begin{align*} (2\vi + \vj) \times \vk =\mathstrut \amp (2\vi \times \vk) + (\vj \times \vk) \\ =\mathstrut \amp 2(\vi \times \vk) + (\vj \times \vk) \\ =\mathstrut \amp 2 \vj + \vi. \end{align*}Using these properties along with Table 1.4.2, find the cross product \(\vu\times\vv\) if \(\vu = 2\vi + 3\vj\) and \(\vv = \vi + \vk\text{.}\)
Verify that the cross product \(\vu\times\vv\) you just found in part (e) is orthogonal to both \(\vu\) and \(\vv\text{.}\)

Consider the vectors \(\vu\) and \(\vv\) in the \(xy\)plane as shown below in Figure 1.4.3.
Explain why \(\vu = \vu\vi\) and \(\vv = \vv\cos(\theta) \vi + \vv\sin(\theta) \vj\text{.}\) Then compute the length of \(\vu\times\vv\text{.}\)
Subsection 1.4.1 Computing the cross product
As we have seen in Preview Activity 1.4.1, the cross product \(\vu\times\vv\) is defined for two vectors \(\vu\) and \(\vv\) in \(\R^3\) and produces another vector in \(\R^3\text{.}\) Using the righthand rule, we saw that
\(\vi\times\vj = \vk\) 
\(\vj\times\vi = \vk\) 
\(\vi\times\vk = \vj\) 
\(\vk\times\vi = \vj\) 
\(\vj\times\vk = \vi\) 
\(\vk\times\vj = \vi\) 
If, in addition, we assume the cross product behaves like we think a product should (e.g., the cross product distributes over vector addition), we can compute the cross product in terms of the components of general vectors to find a formula for the cross product. Doing so we see that
(Like the dot product, the cross product arises in physical applications, e.g., torque, but it is more convenient mathematically to begin from an algebraic perspective.)
The previous calculations lead us to define the cross product of vectors in \(\R^3\) as follows.
Definition 1.4.4.
The cross product \(\vu \times \vv\) of vectors \(\vu = u_1\vi + u_2\vj + u_3\vk\) and \(\vv = v_1\vi + v_2\vj + v_3\vk\) in \(\R^3\) is the vector
At first, this may look intimidating and difficult to remember. However, if we rewrite the expression in Equation (1.4.1) using determinants, important structure emerges. The determinant of a \(2\times2\) matrix is
It follows that we can thus rewrite Equation (1.4.1) in the form
For those familiar with the determinant of a \(3\times3\) matrix, we write the mnemonic as
Activity 1.4.2.
Suppose \(\vu = \langle 0, 1, 3\rangle\) and \(\vv = \langle 2, 1, 0\rangle\text{.}\) Use the formula (1.4.1) for the following.
Find the cross product \(\vu\times\vv\text{.}\)
Evaluate the dot products \(\vu\cdot(\vu\times\vv)\) and \(\vv\cdot(\vu\times\vv)\text{.}\) What does this tell you about the geometric relationship among \(\vu\text{,}\) \(\vv\text{,}\) and \(\vu\times\vv\text{?}\)
Find the cross product \(\vv\times \vi\text{.}\)
Multiplication of real numbers is associative, which means, for instance, that \((2\cdot 5)\cdot 3 = 2\cdot(5\cdot 3)\text{.}\) Is it true that the cross product of vectors is associative? For instance, is it true that \((\vu\times\vv)\times\vi = \vu\times(\vv\times\vi)\text{?}\)
Find the cross product \(\vu\times\vu\text{.}\)
The cross product satisfies the following properties, some of which were illustrated in Preview Activity 1.4.1 and may be easily verified from the definition (1.4.1).
Properties of the cross product.
Let \(\vu\text{,}\) \(\vv\text{,}\) and \(\vw\) be vectors in \(\R^3\text{,}\) and let \(c\) be a scalar. Then
\(\vu \times \vv = (\vv \times \vu)\)
\((\vu +\vv)\times \vw = (\vu \times \vw) + (\vv \times \vw)\)
\((c\vu)\times \vw = c(\vu \times \vw) = \vu \times (c\vv)\)
\(\vu\times\vv = \vzero\) if \(\vu\) and \(\vv\) are parallel.

The cross product is not associative; that is, in general
\begin{equation*} (\vu\times\vv)\times\vw \neq \vu\times(\vv\times\vw). \end{equation*}
Just as we found for the dot product, the cross product provides us with useful geometric information. In particular, both the length and direction of the cross product \(\vu\times\vv\) encode information about the geometric relationship between \(\vu\) and \(\vv\text{.}\)
Subsection 1.4.2 The Length of \(\vu\times\vv\)
We may ask whether the length \(\vu\times\vv\) has any relationship to the lengths of \(\vu\) and \(\vv\text{.}\) To investigate, we will compute the square of the length \(\vu\times\vv^2\) and denote by \(\theta\) the angle between \(\vu\) and \(\vv\text{,}\) as in Section 1.3. Doing so, we find through some significant algebra that
Therefore, we have found \(\vu\times\vv^2 = \vu^2\vv^2\sin^2(\theta)\text{,}\) which means that
Note that the third property stated above says that \(\vu\times\vv = \vzero\) if \(\vu\) and \(\vv\) are parallel. This is reflected in Equation (1.4.2) since \(\sin(\theta)=0\) if \(\vu\) and \(\vv\) are parallel, which implies that \(\vu\times\vv = \vzero\text{.}\)
Equation (1.4.2) also has a geometric implication. Consider the parallelogram formed by two vectors \(\vu\) and \(\vv\text{,}\) as shown in Figure 1.4.5.
Remember that the area of a parallelogram is the product of its base and height. As shown in the figure, we may consider the base of the parallelogram to be \(\vu\) and the height to be \(\vv\sin(\theta)\text{.}\) This means that the area of the parallelogram formed by \(\vu\) and \(\vv\) is
This leads to the following interesting fact.
The length of the cross product.
The length, \(\vu \times \vv\text{,}\) of the cross product of vectors \(\vu\) and \(\vv\) is the area of the parallelogram determined by \(\vu\) and \(\vv\text{.}\)
Note also that if \(\vu = u_1\vi + u_2\vj + 0\vk\) and \(\vv = v_1\vi + v_2\vj + 0\vk\) are vectors in the \(xy\)plane, then Equation (1.4.1) shows that the area of the parallelogram determined by \(\vu\) and \(\vv\) is \(\vu \times \vv = u_1v_2u_2v_1\) is the absolute value of the \(2 \times 2\) determinant \(\left \begin{array}{cc} u_1 \amp u_2 \\ v_1 \amp v_2 \end{array} \right.\) So the absolute value of a determinant of a \(2 \times 2 \) matrix is also the area of a parallelogram.
Activity 1.4.3.
Find the area of the parallelogram formed by the vectors \(\vu = \langle 1,3, 2\rangle\) and \(\vv=\langle 3,0,1\rangle\text{.}\)
Find the area of the parallelogram in \(\R^3\) whose vertices are \((1,0,1)\text{,}\) \((0,0,1)\text{,}\) \((2,1,0)\text{,}\) and \((1,1,0)\text{.}\) (Hint: It might be helpful to draw a picture to see how the vertices are arranged so you can determine which vectors you might use.)
Subsection 1.4.3 The Direction of \(\vu\times\vv\)
Now that we understand the length of \(\vu\times\vv\text{,}\) we will investigate its direction. Remember from Preview Activity 1.4.1 that cross products involving the vectors \(\vi\text{,}\) \(\vj\text{,}\) and \(\vk\) resulted in vectors that are orthogonal to the two terms. We will see that this holds more generally.
We begin by computing \(\vu\cdot(\vu\times\vv)\text{,}\) and see that
To summarize, we have \(\vu\cdot(\vu\times\vv) = 0\text{,}\) which implies that \(\vu\) is orthogonal to \(\vu\times\vv\text{.}\) In the same way, we can show that \(\vv\) is orthogonal to \(\vu\times\vv\text{.}\) The net effect is that \(\vu\times\vv\) is a vector that is perpendicular to both \(\vu\) and \(\vv\text{,}\) and hence \(\vu\times\vv\) is perpendicular to the plane determined by \(\vu\) and \(\vv\text{.}\) Moreover, the direction of \(\vu\times\vv\) is determined by applying the righthand rule to \(\vu\) and \(\vv\text{,}\) as we saw in Preview Activity 1.4.1. In light of our earlier work that showed \(\vu\vv\sin(\theta) = \vu\times\vv.\text{,}\) we may now express \(\vu \times \vv\) in the following different way.
The cross product as normal vector.
Suppose that \(\vu\) and \(\vv\) are not parallel and that \(\vn\) is the unit vector perpendicular to the plane containing \(\vu\) and \(\vv\) determined by the righthand rule. Then
There is yet one more geometric implication we may draw from this result. Suppose \(\vu\text{,}\) \(\vv\text{,}\) and \(\vw\) are vectors in \(\R^3\) that are not coplanar and that form a threedimension parallelepiped as shown in Figure 1.4.6.
The volume of the parallelepiped is determined by multiplying \(A\text{,}\) the area of the base, by the height \(h\text{.}\) As we have just seen, the area of the base is \(\vu\times\vv\text{.}\) Moreover, the height \(h=\vw\cos(\alpha)\) where \(\alpha\) is the angle between \(\vw\) and the vector \(\vn\text{,}\) which is orthogonal to the plane formed by \(\vu\) and \(\vv\text{.}\) Since \(\vn\) is parallel to \(\vu\times\vv\text{,}\) the angle between \(\vw\) and \(\vu\times\vv\) is also \(\alpha\text{.}\) This shows that
and therefore
The cross product and the volume of a parallelepiped.
The volume of the parallelepiped determined by \(\vu\text{,}\) \(\vv\text{,}\) and \(\vw\) is \((\vu\times\vv)\cdot\vw\text{.}\)
As a dot product of two vectors, the quantity \((\vu \times \vv) \cdot \vw\) is a scalar and is called the triple scalar product.
Activity 1.4.4.
Suppose \(\vu = \langle 3, 5, 1\rangle\) and \(\vv = \langle 2, 2, 1\rangle\text{.}\)
Find two unit vectors orthogonal to both \(\vu\) and \(\vv\text{.}\)
Find the volume of the parallelepiped formed by the vectors \(\vu\text{,}\) \(\vv\text{,}\) and \(\vw = \langle 3,3,1\rangle\text{.}\)
Find a vector orthogonal to the plane containing the points \((0,1,2)\text{,}\) \((4,1,0)\text{,}\) and \((2,2,2)\text{.}\)
Given the vectors \(\vu\) and \(\vv\) shown below in Figure 1.4.7, sketch the cross products \(\vu\times\vv\) and \(\vv\times\vu\text{.}\)
Do the vectors \(\va = \langle 1,3,2\rangle\text{,}\)\(\vb=\langle2,1,4\rangle\text{,}\) and \(\vc=\langle 0, 1, 0\rangle\) in standard position lie in the same plane? Use the concepts from this section to explain.
Subsection 1.4.4 Torque is measured by a cross product
We have seen that the cross product enables us to produce a vector perpendicular to two given vectors, to measure the area of a parallelogram, and to measure the volume of a parallelepiped. Besides these geometric applications, the cross product also enables us to describe a physical quantity called torque.
Suppose that we would like to turn a bolt using a wrench as shown in Figure 1.4.8. If a force \(\vF\) is applied to the wrench and \(\vr\) is the vector from the position on the wrench at which the force is applied to center of the bolt, we define the torque, \(\tau\text{,}\) to be
When a force is applied to an object, Newton's Second Law tells us that the force is equal to the rate of change of the object's linear momentum. Similarly, the torque applied to an object is equal to the rate of change of the object's angular momentum. In other words, torque will cause the bolt to rotate.
In many industrial applications, bolts are required to be tightened using a specified torque. Of course, the magnitude of the torque is \(\tau =\vF\times\vr=\vF\vr\sin(\theta)\text{.}\) Thus, to produce a larger torque, we can increase either \(\vF\) or \(\vr\text{,}\) which you may know if you have ever removed lug nuts when changing a flat tire. The ancient Greek mathematician Archimedes said: “Give me a lever long enough and a fulcrum on which to place it, and I shall move the world.” A modern spin on this statement is: “Allow me to make \(\vr\) large enough, and I shall produce a torque large enough to move the world.”
Subsection 1.4.5 Comparing the dot and cross products
Finally, it is worthwhile to compare and contrast the dot and cross products.
\(\vu\cdot\vv\) is a scalar, while \(\vu\times\vv\) is a vector.
\(\vu\cdot\vv = \vv\cdot\vu\text{,}\) while \(\vu\times\vv = \vv\times\vu\)
\(\vu\cdot\vv = \vu\vv\cos(\theta)\text{,}\) while \(\vu\times\vv = \vu\vv\sin(\theta)\text{.}\)
\(\vu\cdot\vv = 0\) if \(\vu\) and \(\vv\) are perpendicular, while \(\vu\times\vv = \vzero\) if \(\vu\) and \(\vv\) are parallel.
Subsection 1.4.6 Summary

The cross product is defined only for vectors in \(\R^3\text{.}\) The cross product of vectors \(\vu = u_1 \vi + u_2 \vj + u_3 \vk\) and \(\vv = v_1 \vi + v_2 \vj + v_3 \vk\) in \(\R^3\) is the vector
\begin{equation*} \vu \times \vv = (u_2v_3u_3v_2) \vi  (u_1v_3  u_3v_1) \vj + (u_1v_2  u_2v_1) \vk. \end{equation*} 
Geometrically, the cross product is
\begin{equation*} \vu \times \vv = \vu \ \vv \ \sin(\theta) \ \vn, \end{equation*}where \(\theta\) is the angle between \(\vu\) and \(\vv\) and \(\vn\) is a unit vector perpendicular to both \(\vu\) and \(\vv\) as determined by the righthand rule.
The cross product of vectors \(\vu\) and \(\vv\) is a vector perpendicular to both \(\vu\) and \(\vv\text{.}\)
The magnitude \(\vu \times \vv\) of the cross product of the vectors \(\vu\) and \(\vv\) gives the area of the parallelogram determined by \(\vu\) and \(\vv\text{.}\) Also, the scalar triple product \((\vu \times \vv) \cdot \vw\) gives the volume of the parallelepiped determined by \(\vu\text{,}\) \(\vv\text{,}\) and \(\vw\text{.}\)
Exercises 1.4.7 Exercises
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11.
Let \(\vu = 2\vi + \vj\) and \(\vv = \vi + 2\vj\) be vectors in \(\R^3\text{.}\)
Without doing any computations, find a unit vector that is orthogonal to both \(\vu\) and \(\vv\text{.}\) What does this tell you about the formula for \(\vu \times \vv\text{?}\)
Using the properties of the cross product and what you know about cross products involving the fundamental vectors \(\vi\) and \(\vj\text{,}\) compute \(\vu \times \vv\text{.}\)
Next, use the determinant version of Equation (1.4.1) to compute \(\vu \times \vv\text{.}\) Write one sentence that compares your results in (a), (b), and (c).
Find the area of the parallelogram determined by \(\vu\) and \(\vv\text{.}\)
12.
Let \(\vx = \langle 1, 1, 1 \rangle\) and \(\vy = \langle 0, 3, 2 \rangle\text{.}\)
Are \(\vx\) and \(\vy\) orthogonal? Are \(\vx\) and \(\vy\) parallel? Clearly explain how you know, using appropriate vector products.
Find a unit vector that is orthogonal to both \(\vx\) and \(\vy\text{.}\)
Express \(\vy\) as the sum of two vectors: one parallel to \(\vx\text{,}\) the other orthogonal to \(\vx\text{.}\)
Determine the area of the parallelogram formed by \(\vx\) and \(\vy\text{.}\)
13.
Consider the triangle in \(\R^3\) formed by \(P(3, 2, 1)\text{,}\) \(Q(1, 2, 4)\text{,}\) and \(R(4, 4, 0)\text{.}\)
Find \(\overrightarrow{PQ}\) and \(\overrightarrow{PR}\text{.}\)
Observe that the area of \(\triangle PQR\) is half of the area of the parallelogram formed by \(\overrightarrow{PQ}\) and \(\overrightarrow{PR}\text{.}\) Hence find the area of \(\triangle PQR\text{.}\)
Find a unit vector that is orthogonal to the plane that contains points \(P\text{,}\) \(Q\text{,}\) and \(R\text{.}\)
Determine the measure of \(\angle PQR\text{.}\)
14.
One of the properties of the cross product is that \((\vu+\vv) \times \vw = (\vu \times \vw) + (\vv \times \vw)\text{.}\) That is, the cross product distributes over vector addition on the right. Here we investigate whether the cross product distributes over vector addition on the left.

Let \(\vu = \langle 1,2,1 \rangle\text{,}\) \(\vv = \langle 4,3,6 \rangle\text{,}\) and \(\vv = \langle 4,7,2 \rangle\text{.}\) Calculate
\begin{equation*} \vu \times (\vv + \vw) \ \ \text{ and } \ \ (\vu \times \vv) + (\vu \times \vw). \end{equation*}What do you notice?

Use the properties of the cross product to show that in general
\begin{equation*} \vx \times (\vy + \vz) = (\vx \times \vy) + (\vx \times \vz) \end{equation*}for any vectors \(\vx\text{,}\) \(\vy\text{,}\) and \(\vz\) in \(\R^3\text{.}\)
15.
Let \(\vu = \langle u_1, u_2, u_3 \rangle\text{,}\) \(\vv = \langle v_1, v_2, v_3 \rangle\text{,}\) and \(\vw = \langle w_1, w_2, w_3 \rangle\) be vectors in \(\R^3\text{.}\) In this exercise we investigate properties of the triple scalar product \((\vu \times \vv) \cdot \vw\text{.}\)
Show that \((\vu \times \vv) \cdot \vw = \left\begin{array}{ccc} u_1 \amp u_2 \amp u_3 \\ v_1 \amp v_2 \amp v_3 \\ w_1 \amp w_2 \amp w_3 \end{array} \right \text{.}\)
Show that \(\left\begin{array}{ccc} u_1 \amp u_2 \amp u_3 \\ v_1 \amp v_2 \amp v_3 \\ w_1 \amp w_2 \amp w_3 \end{array} \right = \left\begin{array}{ccc} v_1 \amp v_2 \amp v_3 \\ u_1 \amp u_2 \amp u_3 \\ w_1 \amp w_2 \amp w_3 \end{array} \right\text{.}\) Conclude that interchanging the first two rows in a \(3 \times 3\) matrix changes the sign of the determinant. In general (although we won't show it here), interchanging any two rows in a \(3 \times 3\) matrix changes the sign of the determinant.

Use the results of parts (a) and (b) to argue that
\begin{equation*} (\vu \times \vv) \cdot \vw = (\vw \times \vu) \cdot \vv = (\vv \times \vw) \cdot \vu. \end{equation*} 
Now suppose that \(\vu\text{,}\) \(\vv\text{,}\) and \(\vw\) do not lie in a plane when they eminate from a common initial point.
Given that the parallepiped determined by \(\vu\text{,}\) \(\vv\text{,}\) and \(\vw\) must have positive volume, what can we say about \((\vu \times \vv) \cdot \vw\text{?}\)
Now suppose that \(\vu\text{,}\) \(\vv\text{,}\) and \(\vw\) all lie in the same plane. What value must \((\vu \times \vv) \cdot \vw\) have? Why?
Explain how (i.) and (ii.) show that if \(\vu\text{,}\) \(\vv\text{,}\) and \(\vw\) all eminate from the same initial point, then \(\vu\text{,}\) \(\vv\text{,}\) and \(\vw\) lie in the same plane if and only if \((\vu \times \vv) \cdot \vw = 0\text{.}\) This provides a straightforward computational method for determining when three vectors are coplanar.