Vectors in Space

# 11 The Cross Product

### Learning Objectives

• Calculate the cross product of two given vectors.
• Use determinants to calculate a cross product.
• Find a vector orthogonal to two given vectors.
• Determine areas and volumes by using the cross product.
• Calculate the torque of a given force and position vector.

Imagine a mechanic turning a wrench to tighten a bolt. The mechanic applies a force at the end of the wrench. This creates rotation, or torque, which tightens the bolt. We can use vectors to represent the force applied by the mechanic, and the distance (radius) from the bolt to the end of the wrench. Then, we can represent torque by a vector oriented along the axis of rotation. Note that the torque vector is orthogonal to both the force vector and the radius vector.

In this section, we develop an operation called the cross product, which allows us to find a vector orthogonal to two given vectors. Calculating torque is an important application of cross products, and we examine torque in more detail later in the section.

### The Cross Product and Its Properties

The dot product is a multiplication of two vectors that results in a scalar. In this section, we introduce a product of two vectors that generates a third vector orthogonal to the first two. Consider how we might find such a vector. Let and be nonzero vectors. We want to find a vector orthogonal to both and —that is, we want to find such that and Therefore, and must satisfy

If we multiply the top equation by and the bottom equation by and subtract, we can eliminate the variable which gives

If we select

we get a possible solution vector. Substituting these values back into the original equations gives

That is, vector

is orthogonal to both and which leads us to define the following operation, called the cross product.

Definition

Let Then, the cross product is vector

From the way we have developed it should be clear that the cross product is orthogonal to both and However, it never hurts to check. To show that is orthogonal to we calculate the dot product of and

In a similar manner, we can show that the cross product is also orthogonal to

Finding a Cross Product

Let ((Figure)). Find

Finding a cross product to two given vectors.

Substitute the components of the vectors into (Figure):

Find for and Express the answer using standard unit vectors.

Hint

Use the formula

Although it may not be obvious from (Figure), the direction of is given by the right-hand rule. If we hold the right hand out with the fingers pointing in the direction of then curl the fingers toward vector the thumb points in the direction of the cross product, as shown.

The direction of is determined by the right-hand rule.

Notice what this means for the direction of If we apply the right-hand rule to we start with our fingers pointed in the direction of then curl our fingers toward the vector In this case, the thumb points in the opposite direction of (Try it!)

Anticommutativity of the Cross Product

Let and Calculate and and graph them.

Are the cross products and in the same direction?

We have

We see that, in this case, ((Figure)). We prove this in general later in this section.

The cross products and are both orthogonal to and but in opposite directions.

Suppose vectors and lie in the xy-plane (the z-component of each vector is zero). Now suppose the x– and y-components of and the y-component of are all positive, whereas the x-component of is negative. Assuming the coordinate axes are oriented in the usual positions, in which direction does point?

Up (the positive z-direction)

Hint

Remember the right-hand rule.

The cross products of the standard unit vectors and can be useful for simplifying some calculations, so let’s consider these cross products. A straightforward application of the definition shows that

(The cross product of two vectors is a vector, so each of these products results in the zero vector, not the scalar It’s up to you to verify the calculations on your own.

Furthermore, because the cross product of two vectors is orthogonal to each of these vectors, we know that the cross product of and is parallel to Similarly, the vector product of and is parallel to and the vector product of and is parallel to We can use the right-hand rule to determine the direction of each product. Then we have

These formulas come in handy later.

Cross Product of Standard Unit Vectors

Find

We know that Therefore,

Find

Hint

Remember the right-hand rule.

As we have seen, the dot product is often called the scalar product because it results in a scalar. The cross product results in a vector, so it is sometimes called the vector product. These operations are both versions of vector multiplication, but they have very different properties and applications. Let’s explore some properties of the cross product. We prove only a few of them. Proofs of the other properties are left as exercises.

Properties of the Cross Product

Let and be vectors in space, and let be a scalar.

#### Proof

For property we want to show We have

Unlike most operations we’ve seen, the cross product is not commutative. This makes sense if we think about the right-hand rule.

For property this follows directly from the definition of the cross product. We have

Then, by property i., as well. Remember that the dot product of a vector and the zero vector is the scalar whereas the cross product of a vector with the zero vector is the vector

Property looks like the associative property, but note the change in operations:

Using the Properties of the Cross Product

Use the cross product properties to calculate

Use the properties of the cross product to calculate

Hint

So far in this section, we have been concerned with the direction of the vector but we have not discussed its magnitude. It turns out there is a simple expression for the magnitude of involving the magnitudes of and and the sine of the angle between them.

Magnitude of the Cross Product

Let and be vectors, and let be the angle between them. Then,

#### Proof

Let and be vectors, and let denote the angle between them. Then

Taking square roots and noting that for we have the desired result:

This definition of the cross product allows us to visualize or interpret the product geometrically. It is clear, for example, that the cross product is defined only for vectors in three dimensions, not for vectors in two dimensions. In two dimensions, it is impossible to generate a vector simultaneously orthogonal to two nonparallel vectors.

Calculating the Cross Product

Use (Figure) to find the magnitude of the cross product of and

We have

Use (Figure) to find the magnitude of where and

Hint

Vectors and are orthogonal.

### Determinants and the Cross Product

Using (Figure) to find the cross product of two vectors is straightforward, and it presents the cross product in the useful component form. The formula, however, is complicated and difficult to remember. Fortunately, we have an alternative. We can calculate the cross product of two vectors using determinant notation.

A determinant is defined by

For example,

A determinant is defined in terms of determinants as follows:

(Figure) is referred to as the expansion of the determinant along the first row. Notice that the multipliers of each of the determinants on the right side of this expression are the entries in the first row of the determinant. Furthermore, each of the determinants contains the entries from the determinant that would remain if you crossed out the row and column containing the multiplier. Thus, for the first term on the right, is the multiplier, and the determinant contains the entries that remain if you cross out the first row and first column of the determinant. Similarly, for the second term, the multiplier is and the determinant contains the entries that remain if you cross out the first row and second column of the determinant. Notice, however, that the coefficient of the second term is negative. The third term can be calculated in similar fashion.

Using Expansion Along the First Row to Compute a Determinant

Evaluate the determinant

We have

Evaluate the determinant

Hint

Expand along the first row. Don’t forget the second term is negative!

Technically, determinants are defined only in terms of arrays of real numbers. However, the determinant notation provides a useful mnemonic device for the cross product formula.

Rule: Cross Product Calculated by a Determinant

Let and be vectors. Then the cross product is given by

Using Determinant Notation to find

Let and Find

We set up our determinant by putting the standard unit vectors across the first row, the components of in the second row, and the components of in the third row. Then, we have

Notice that this answer confirms the calculation of the cross product in (Figure).

Use determinant notation to find where and

Hint

Calculate the determinant

### Using the Cross Product

The cross product is very useful for several types of calculations, including finding a vector orthogonal to two given vectors, computing areas of triangles and parallelograms, and even determining the volume of the three-dimensional geometric shape made of parallelograms known as a parallelepiped. The following examples illustrate these calculations.

Finding a Unit Vector Orthogonal to Two Given Vectors

Let and Find a unit vector orthogonal to both and

The cross product is orthogonal to both vectors and We can calculate it with a determinant:

Normalize this vector to find a unit vector in the same direction:

Thus, is a unit vector orthogonal to and

Find a unit vector orthogonal to both and where and

Hint

Normalize the cross product.

To use the cross product for calculating areas, we state and prove the following theorem.

Area of a Parallelogram

If we locate vectors and such that they form adjacent sides of a parallelogram, then the area of the parallelogram is given by ((Figure)).

The parallelogram with adjacent sides and has base and height

#### Proof

We show that the magnitude of the cross product is equal to the base times height of the parallelogram.

Finding the Area of a Triangle

Let be the vertices of a triangle ((Figure)). Find its area.

Finding the area of a triangle by using the cross product.

We have and The area of the parallelogram with adjacent sides and is given by

The area of is half the area of the parallelogram, or

Find the area of the parallelogram with vertices and

Hint

Sketch the parallelogram and identify two vectors that form adjacent sides of the parallelogram.

### The Triple Scalar Product

Because the cross product of two vectors is a vector, it is possible to combine the dot product and the cross product. The dot product of a vector with the cross product of two other vectors is called the triple scalar product because the result is a scalar.

Definition

The triple scalar product of vectors and is

Calculating a Triple Scalar Product

The triple scalar product of vectors and is the determinant of the matrix formed by the components of the vectors:

#### Proof

The calculation is straightforward.

Calculating the Triple Scalar Product

Let Calculate the triple scalar product

Apply (Figure) directly:

Calculate the triple scalar product where and

Hint

Place the vectors as the rows of a matrix, then calculate the determinant.

When we create a matrix from three vectors, we must be careful about the order in which we list the vectors. If we list them in a matrix in one order and then rearrange the rows, the absolute value of the determinant remains unchanged. However, each time two rows switch places, the determinant changes sign:

Verifying this fact is straightforward, but rather messy. Let’s take a look at this with an example:

Switching the top two rows we have

Rearranging vectors in the triple products is equivalent to reordering the rows in the matrix of the determinant. Let and Applying (Figure), we have

We can obtain the determinant for calculating by switching the bottom two rows of Therefore,

Following this reasoning and exploring the different ways we can interchange variables in the triple scalar product lead to the following identities:

Let and be two vectors in standard position. If and are not scalar multiples of each other, then these vectors form adjacent sides of a parallelogram. We saw in (Figure) that the area of this parallelogram is Now suppose we add a third vector that does not lie in the same plane as and but still shares the same initial point. Then these vectors form three edges of a parallelepiped, a three-dimensional prism with six faces that are each parallelograms, as shown in (Figure). The volume of this prism is the product of the figure’s height and the area of its base. The triple scalar product of and provides a simple method for calculating the volume of the parallelepiped defined by these vectors.

Volume of a Parallelepiped

The volume of a parallelepiped with adjacent edges given by the vectors is the absolute value of the triple scalar product:

See (Figure).

Note that, as the name indicates, the triple scalar product produces a scalar. The volume formula just presented uses the absolute value of a scalar quantity.

The height of the parallelepiped is given by

#### Proof

The area of the base of the parallelepiped is given by The height of the figure is given by The volume of the parallelepiped is the product of the height and the area of the base, so we have

Calculating the Volume of a Parallelepiped

Let Find the volume of the parallelepiped with adjacent edges ((Figure)).

We have

Thus, the volume of the parallelepiped is units3.

Find the volume of the parallelepiped formed by the vectors and

units3

Hint

Calculate the triple scalar product by finding a determinant.

### Applications of the Cross Product

The cross product appears in many practical applications in mathematics, physics, and engineering. Let’s examine some of these applications here, including the idea of torque, with which we began this section. Other applications show up in later chapters, particularly in our study of vector fields such as gravitational and electromagnetic fields (Introduction to Vector Calculus).

Using the Triple Scalar Product

Use the triple scalar product to show that vectors are coplanar—that is, show that these vectors lie in the same plane.

Start by calculating the triple scalar product to find the volume of the parallelepiped defined by

The volume of the parallelepiped is units3, so one of the dimensions must be zero. Therefore, the three vectors all lie in the same plane.

Are the vectors and coplanar?

No, the triple scalar product is so the three vectors form the adjacent edges of a parallelepiped. They are not coplanar.

Hint

Calculate the triple scalar product.

Finding an Orthogonal Vector

Only a single plane can pass through any set of three noncolinear points. Find a vector orthogonal to the plane containing points and

The plane must contain vectors and

The cross product produces a vector orthogonal to both and Therefore, the cross product is orthogonal to the plane that contains these two vectors:

We have seen how to use the triple scalar product and how to find a vector orthogonal to a plane. Now we apply the cross product to real-world situations.

Sometimes a force causes an object to rotate. For example, turning a screwdriver or a wrench creates this kind of rotational effect, called torque.

Definition

Torque, (the Greek letter tau), measures the tendency of a force to produce rotation about an axis of rotation. Let be a vector with an initial point located on the axis of rotation and with a terminal point located at the point where the force is applied, and let vector represent the force. Then torque is equal to the cross product of and

See (Figure).

Torque measures how a force causes an object to rotate.

Think about using a wrench to tighten a bolt. The torque applied to the bolt depends on how hard we push the wrench (force) and how far up the handle we apply the force (distance). The torque increases with a greater force on the wrench at a greater distance from the bolt. Common units of torque are the newton-meter or foot-pound. Although torque is dimensionally equivalent to work (it has the same units), the two concepts are distinct. Torque is used specifically in the context of rotation, whereas work typically involves motion along a line.

Evaluating Torque

A bolt is tightened by applying a force of N to a 0.15-m wrench ((Figure)). The angle between the wrench and the force vector is Find the magnitude of the torque about the center of the bolt. Round the answer to two decimal places.

Torque describes the twisting action of the wrench.

Substitute the given information into the equation defining torque:

Calculate the force required to produce torque at an angle of from a 150-cm rod.

N

Hint

and

### Key Concepts

• The cross product of two vectors and is a vector orthogonal to both and Its length is given by where is the angle between and Its direction is given by the right-hand rule.
• The algebraic formula for calculating the cross product of two vectors,
is
• The cross product satisfies the following properties for vectors and scalar
• The cross product of vectors and is the determinant
• If vectors and form adjacent sides of a parallelogram, then the area of the parallelogram is given by
• The triple scalar product of vectors and is
• The volume of a parallelepiped with adjacent edges given by vectors is
• If the triple scalar product of vectors is zero, then the vectors are coplanar. The converse is also true: If the vectors are coplanar, then their triple scalar product is zero.
• The cross product can be used to identify a vector orthogonal to two given vectors or to a plane.
• Torque measures the tendency of a force to produce rotation about an axis of rotation. If force is acting at a distance from the axis, then torque is equal to the cross product of and

### Key Equations

• The cross product of two vectors in terms of the unit vectors

For the following exercises, the vectors and are given.

1. Find the cross product of the vectors and Express the answer in component form.
2. Sketch the vectors and

a.
b.

a.
b.

Simplify

Simplify

In the following exercises, vectors and are given. Find unit vector in the direction of the cross product vector Express your answer using standard unit vectors.

where and

where and

Determine the real number such that and are orthogonal, where and

Show that and cannot be orthogonal for any real number, where and

Show that is orthogonal to and where and are nonzero vectors.

Show that is orthogonal to where and are nonzero vectors.

Calculate the determinant

Calculate the determinant

For the following exercises, the vectors and are given. Use determinant notation to find vector orthogonal to vectors and

where is a real number

where is a nonzero real number

Find vector where and

Find vector where and

[T] Use the cross product to find the acute angle between vectors and where and Express the answer in degrees rounded to the nearest integer.

[T] Use the cross product to find the obtuse angle between vectors and where and Express the answer in degrees rounded to the nearest integer.

Use the sine and cosine of the angle between two nonzero vectors and to prove Lagrange’s identity:

Verify Lagrange’s identity for vectors and

Nonzero vectors and are called collinear if there exists a nonzero scalar such that Show that and are collinear if and only if

Nonzero vectors and are called collinear if there exists a nonzero scalar such that Show that vectors and are collinear, where and

Find the area of the parallelogram with adjacent sides and

Find the area of the parallelogram with adjacent sides and

Consider points and

1. Find the area of parallelogram with adjacent sides and
2. Find the area of triangle
3. Find the distance from point to line

a. b. c.

Consider points and

1. Find the area of parallelogram with adjacent sides and
2. Find the area of triangle
3. Find the distance from point to line

In the following exercises, vectors are given.

1. Find the triple scalar product
2. Find the volume of the parallelepiped with the adjacent edges

and

a. b.

and

Calculate the triple scalar products and where and

Calculate the triple scalar products and where and

Find vectors with a triple scalar product given by the determinant

Determine their triple scalar product.

The triple scalar product of vectors is given by the determinant

Find vector

Consider the parallelepiped with edges and where and

1. Find the real number such that the volume of the parallelepiped is units3.
2. For find the height from vertex of the parallelepiped. Sketch the parallelepiped.

a. b.

Consider points and with and positive real numbers.

1. Determine the volume of the parallelepiped with adjacent sides and
2. Find the volume of the tetrahedron with vertices (Hint: The volume of the tetrahedron is of the volume of the parallelepiped.)
3. Find the distance from the origin to the plane determined by Sketch the parallelepiped and tetrahedron.

Let be three-dimensional vectors and be a real number. Prove the following properties of the cross product.

Show that vectors and satisfy the following properties of the cross product.

Nonzero vectors are said to be linearly dependent if one of the vectors is a linear combination of the other two. For instance, there exist two nonzero real numbers and such that Otherwise, the vectors are called linearly independent. Show that are coplanar if and only if they are linear dependent.

Consider vectors and

1. Show that are coplanar by using their triple scalar product
2. Show that are coplanar, using the definition that there exist two nonzero real numbers and such that
3. Show that are linearly independent—that is, none of the vectors is a linear combination of the other two.

Consider points and Are vectors and linearly dependent (that is, one of the vectors is a linear combination of the other two)?

Yes, where and

Show that vectors and are linearly independent—that is, there exist two nonzero real numbers and such that

Let and be two-dimensional vectors. The cross product of vectors and is not defined. However, if the vectors are regarded as the three-dimensional vectors and respectively, then, in this case, we can define the cross product of and In particular, in determinant notation, the cross product of and is given by

Use this result to compute where is a real number.

Consider points and

1. Find the area of triangle
2. Determine the distance from point to the line passing through

Determine a vector of magnitude perpendicular to the plane passing through the x-axis and point

Determine a unit vector perpendicular to the plane passing through the z-axis and point

Consider and two three-dimensional vectors. If the magnitude of the cross product vector is times larger than the magnitude of vector show that the magnitude of is greater than or equal to where is a natural number.

[T] Assume that the magnitudes of two nonzero vectors and are known. The function defines the magnitude of the cross product vector where is the angle between

1. Graph the function
2. Find the absolute minimum and maximum of function Interpret the results.
3. If and find the angle between if the magnitude of their cross product vector is equal to

Find all vectors that satisfy the equation

where is any real number

Solve the equation where is a nonzero vector with a magnitude of

[T] A mechanic uses a 12-in. wrench to turn a bolt. The wrench makes a angle with the horizontal. If the mechanic applies a vertical force of lb on the wrench handle, what is the magnitude of the torque at point (see the following figure)? Express the answer in foot-pounds rounded to two decimal places.

8.66 ft-lb

[T] A boy applies the brakes on a bicycle by applying a downward force of lb on the pedal when the 6-in. crank makes a angle with the horizontal (see the following figure). Find the torque at point Express your answer in foot-pounds rounded to two decimal places.

[T] Find the magnitude of the force that needs to be applied to the end of a 20-cm wrench located on the positive direction of the y-axis if the force is applied in the direction and it produces a N·m torque to the bolt located at the origin.

250 N

[T] What is the magnitude of the force required to be applied to the end of a 1-ft wrench at an angle of to produce a torque of N·m?

[T] The force vector acting on a proton with an electric charge of (in coulombs) moving in a magnetic field where the velocity vector is given by (here, is expressed in meters per second, is in tesla [T], and is in newtons [N]). Find the force that acts on a proton that moves in the xy-plane at velocity (in meters per second) in a magnetic field given by

[T] The force vector acting on a proton with an electric charge of moving in a magnetic field where the velocity vector v is given by (here, is expressed in meters per second, in and in If the magnitude of force acting on a proton is N and the proton is moving at the speed of 300 m/sec in magnetic field of magnitude 2.4 T, find the angle between velocity vector of the proton and magnetic field Express the answer in degrees rounded to the nearest integer.

[T] Consider the position vector of a particle at time where the components of are expressed in centimeters and time in seconds. Let be the position vector of the particle after sec.

1. Determine unit vector (called the binormal unit vector) that has the direction of cross product vector where and are the instantaneous velocity vector and, respectively, the acceleration vector of the particle after seconds.
2. Use a CAS to visualize vectors and as vectors starting at point along with the path of the particle.

a.
b.

A solar panel is mounted on the roof of a house. The panel may be regarded as positioned at the points of coordinates (in meters) and (see the following figure).

1. Find vector perpendicular to the surface of the solar panels. Express the answer using standard unit vectors.
2. Assume unit vector points toward the Sun at a particular time of the day and the flow of solar energy is (in watts per square meter []). Find the predicted amount of electrical power the panel can produce, which is given by the dot product of vectors and (expressed in watts).
3. Determine the angle of elevation of the Sun above the solar panel. Express the answer in degrees rounded to the nearest whole number. (Hint: The angle between vectors and and the angle of elevation are complementary.)

### Glossary

cross product
where and
determinant
a real number associated with a square matrix
parallelepiped
a three-dimensional prism with six faces that are parallelograms
torque
the effect of a force that causes an object to rotate
triple scalar product
the dot product of a vector with the cross product of two other vectors:
vector product
the cross product of two vectors