Area, Volume, and Torque
in Three Dimensions

This is a long module. You may want to go directly to one of the following topics.

• The area of a parallelogram and torque in R³
• The geometric or physical definition of the cross product
• The algebraic definition of the cross product
• The determinant of a 3 by 3 matrix
• Computing the cross product as a determinant
• Algebraic properties of the cross product
• Volume of a parallelepiped in R³
• The triple product

In R³ we often use the following notation

• i for the vector (1, 0, 0)
• j for the vector (0, 1, 0)
• k for the vector (0, 0,1).

Using this notation any vector can be written in the form

We begin with the same two problems we discussed in the previous module in R² except now we look at these problems in R³.

• Find the area of a parallelogram whose sides are determined by the two vectors x and y.

• Find the torque produced by a force represented by the vector y acting on the end of a rod represented by the vector y.

Area and Torque

In R³ as in R² the same formula

where t is the angle from the vector x to the vector y determines the (unsigned) area of the parallelogram and the strength of the torque. This, however, is only part of the story. In R² there are only two kinds of torque -- clockwise and counterclockwise -- but in R³ there are many ways in which a rod with one end fixed at the origin can pivot about the origin. The figure below shows two examples.

We need to do three things to study what happens when a force acts on a rod in R³.

• Find the strength of the torque.

• Find the "axle" around which the rod pivots or rotates.

• Find the direction of rotation around the "axle."

where t is the angle from x to y and it is clear that the axle around which the rod pivots will be perpendicular to the rod x and to the force y.

The Geometrical Definition of the Cross Product

We will define a new operation on vectors -- called the cross product or wedge product and written

or

that gives us all the information we need about the resulting torque when the force y acts on the rod x.

The strength of the torque will be the magnitude of the vector x ^ y. Thus, ||x ^ y|| = ||x|| ||y|| |sin t| where t is the angle from the vector x to the vector y. The vector x ^ y will be perpendicular to the vector x and to the vector y. The rod will appear to pivot counterclockwise around the axle x ^ y when viewed from a vantage point at the end of the vector x ^ y as shown in the figure at the right.

Notice that the first two properties listed above determine how long the vector x ^ y is and a direction parallel to x ^ y but don't by themselves determine the direction in which x ^ y is pointing. For example, the first two properties imply that i ^ j = k or that i ^ j = -k. The third property resolves this ambiguity. i ^ j = k.

There are several other ways that this last property can be expressed.

• The right-hand rule

• The right-hand screw rule

Use the three properties above to determine each of the following. Your CAS window contains a procedure that computes the cross product algebraically. You can use this procedure to check your answers.

1. i ^ k

2. j ^ k

3. (i + j) ^ j

4. (i + j) ^ k

5. (a, b, 0) ^ (u, v, 0)

6. (a, 0, c) ^ (u, 0, w)

7. (0, b, c) ^ (0, v, w)

The Algebraic Definition of the Cross Product

Your work above should convince you of two things.

• It is possible to compute the cross product using the three properties above.

• it is tedious to do so.

In the remainder of this module we develop an algebraic formula for computing the cross product.

To find an algebraic formula for the cross product we need to look at our three dimensional world. The best way to do this is to use both a Java applet and a physical model -- for example, a TinkerToy model like the one shown in the picture below.

The connector in the center of TinkerToy model represents the origin. Each of the three rods represents one of the three axes. You should label the axes -- x, y, and z -- so that you can keep track of which axis is which.

If you don't have such a physical model then you can work with just the Java applet. You will need to refer to refer to the TinkerToy model and the Java applet several times. Click here to open a new window with the Java applet. Arrange the new window so that you can move easily between these two windows by clicking on the inactive window to make it active.

When the Java applet appears you should see the picture below.

The most noticeable features in this picture are the origin and the three axes.

• The origin is marked by a large black ball.

• The x-axis is marked by a red rod with a red ball on the end.

• The y-axis is marked by a blue rod with a blue ball on the end.

• The z-axis is marked by a magenta or purple rod with a magenta or purple ball on the end.

Finally, there is a black vector with one end at the end of the green rod and the other end marked by a small black ball. This vector represents a force acting on the end of the green rod. We denote this vector by F for force. We are interested in the vector

that represents the torque. We will attempt to determine each of the three components p, q, and r, of the torque by looking at this picture from each of the following vantage points.

• The component p represents the torque as viewed from a vantage point from far away in the direction i.

• The component q represents the torque as viewed from a vantage point from far away in the direction j.

• The component r represents the torque as viewed from a vantage point from far away in the direction k.

You can look at this situation from different vantage points physically by holding the TinkerToy object in different positions -- for example, the picture below shows the TinkerToy object from a vantage point far away along the positive y-axis -- that is, far away from the direction of the vector j. This is the vantage point that is appropriate for determining the portion of the torque around the y-axis -- that is, the component q of the torque.

You can move set up in the Java applet around as follows. Click close to any one of the three balls on the ends of the three axes. When you do so the set up will rotate so that this ball moves to the point at which you clicked. You can rotate the set up any way that you want by dragging the balls on the end of the axes or by a series of clicks.

We will eventually want to move the set up so that each (positive) axis in turn points at you. When an axis is pointing directly at you, the abll on the end of the axis will be centered in front of the origin and a colored bar will appear next to the axis name at the bottom of the applet. We need to be aware of when an axis s pointing into the screen rather than out of the screen. When this happens a black bar will appear to the right of the axis name at the bottom of the applet.

Following the instructions above, rotate the set up so that it looks like the picture below -- with the z-axis pointing directly toward you; the x-axis pointing to your right; and the y-axis pointing upward. This is the position we need to look at the p component of the torque. Notice that the purple ball at the end of the z-axis is centered in the black ball representing the origin because in this position the z-axis is lined up with the origin. Notice the magenta bar next to the word z-axis at the bottom of the display. This bar indicates that the positive end of the z-axis is pointing toward you.

Using the notation

We compute the three components of the torque as follows. Click on each formula for an explanation of that formula.

• p = b w - c v

• q = c u - a w

• r = a v - b u

Putting this all together we get the following algebraic formula for the cross product.

or

You may have already seen the determinant of a 3 by 3 matrix, defined below.

Definition

The determinant of a 3 by 3 matrix

written, det(A), or

is defined by

Using the determinant the cross product of two vectors in R³ can be written

Compute the following cross products using the algebraic formula developed above. Check your work using the cross product procedure in your CAS window.

1. (1, 2, 3) ^ (4, 5, 6)

2. (2, 0, 4) ^ (4, 2, -1)

3. (4, 2, -5) ^ (-1, -2, 4)

4. (4, -1, -1) ^ (2, -2, -3)

5. (1, 2, 3) ^ (2, 4, 6). Comment on your answer.

6. Prove that if
   

   then

   

The following theorem lists some properties of the cross product in R³. It is easy to prove these properties using the algebraic formula for the cross product.

Theorem

Suppose that x, y, and z are vectors in R³ and that c is a real number. Then

• x ^ y = - (y ^ x)

• (c x) ^ y = c (x ^ y) = x ^ (c y)

• (x + y) ^ z = (x ^ z) + (y ^ z)

• x ^ (y + z) = (x ^ y) + (x ^ z)

• O ^ x = x ^ O = O

• x ^ x = O

The next theorem verifies that using the algebraic formula for the cross product we get

where t is the angle from x to y as expected.

Theorem

where t is the angle from x to y.

Proof

Let

The angle t can be found using

Thus

We want to show that

This is easy to show using a straightforward but tedious calculation. One way to do it is using your CAS window.

The next theorem verifies that

using the algebraic formula for the cross product.

Theorem

Proof: The proof is straightforward computation and can be done using your CAS window.

The Volume of a Parallelepiped in R³

The figure below shows a parallelogram in R³.

The next figure shows a parallelpiped. The base of this parallelepiped is the same papallelogram as in the figure above. By our work above the area of this parallelogram is ||x ^ y||.

The volume of a parallelepiped is the area of its base mutliplied by its height, h, as shown in the figure below.

The height h is the dot product

where u is a unit vector perpendicular to the base -- that is, to the parallelogram whose sides are x and y. Therefore,

and the volume of the parallelepiped is

The product

is sometimes called the triple product and is sometimes written x ^ y ^ z. In view of our work above it can be computed by the formula

The triple product gives us the (signed) volume of the parallelepiped. Its absolute value is the volume of the parallelepiped and its sign is

• Positive if using your right hand you can line your thumb up with the vector x, your index finger with the vector y, and your middle finger up with the vector z.

• Negative if using your left hand you can line your thumb up with the vector x, your index finger with the vector y, and your middle finger up with the vector z.

Compare the following theorem with its two-dimensional analog.

Theorem

• 
  x ^ y ^ z = z ^ x ^ y
            = y ^ z ^ x
            = -(x ^ z ^ y)
            = -(z ^ y ^ x)
            = -(y ^ x ^ z)
  

• (cx) ^ y ^ z = x ^ (cy) ^ z = x ^ y ^ (cz) = c (x ^ y ^ z)

• 
  (u + v) ^ y ^ z = (u ^ y ^ z) + (v ^ y ^ z)
  x ^ (u + v) ^ z = (x ^ u ^ z  + (x ^ v ^ z)
  x ^ y ^ (u + v) = (x ^ y ^ u) + (x ^ y ^ v)
  

• O ^ x ^ y = x ^ O ^ z = x ^ y ^ O = O

• x ^ x ^ z = x ^ y ^ y = z ^ y ^ z = O

Proof

It is a good exercise to prove the statements in the theorem above algebraically and then explain each one geometrically.