1

1.1

Cross Product

Definitions

Unlike the dot product, the cross product is only defined for 3-D vectors. In this section, when we use the word vector, we will mean 3-D vector. Definition 1 (cross product) The cross product also called vector product of two vectors u = ux, uy , uz u × v , is defined to be and v = vx, vy , vz , denoted ⎛

⎞

⎛

⎞

⎛

⎞

ux vx uy vz − uz vy ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ uy ⎠ × ⎝ vy ⎠ = ⎝ uz vx − uxvz ⎠ uz vz uxvy − uy vx

Thus, the cross product of two 3-D vectors is also a 3-D vector. This formula is not easy to remember. However, if you know about matrices and the determinant of a matrix, the cross product can be expressed in term of them. Let us first quickly review what they are.

Definition 2 We only give the definition of the determinant of a 2 × 2 and a 3 × 3 matrix.

1. The determinant of a 2 × 2 matrix

a b

c d

is defined to be

a b

c d

a b , denoted by c d

= ad − bc

⎡

a1 ⎢ 2. The determinant of a 3×3 matrix ⎣ b1 c1

a a a

1 2 3

by b1 b2 b3 is defined to be

c1 c2 c3

a a a

1

b b

b 2 3

2 3

b1 b2 b3 = a1

−a2 1

c2 c3

c1

c1 c2 c3

⎤

a2 a3 ⎥ b2 b3 ⎦denoted c2 c3

b3 c3

+a3

b1 b2 c1 c2

1 2

Example 3 Find

7 3

1 2 3

Example 4 Find 3 1 1

4 7 2

Proposition 5 If u = ux, uy , uz and v = vx, vy , vz then

− →

→ −

→ − j k

i

u × v = ux uy uz

vx vy vz

Which makes it much easier to remember. − − Example 6 For → u = 3, 1, 1 and → v = 4, 7, 2, compute u × v. The above tells us how to compute the cross product. However, it does not tell us what the cross product represents. There is a very nice geometric interpretation of the interpretation of the cross product.

1.2

Properties

Theorem 7 Let u and v denote two non-zero vectors. Then, the following is true: 1. u× u = 0 2. u × v is perpendicular to both u and v. 3. u × v = u v sin α where α is the smallest angle between u and v (0 ≤ α ≤ π). → → 4. u ×v = u v sin α− n where − n is the unit vector perpendicular to both u and v whose direction is determined by the righ-hand rule.

Remark 8 The above properties tell us that u × v is the vector perpendicular to both u and v which direction is given by the right-hand rule and whose magnitude is u v sin α. This is very important. There are many situations in which one needs to find a vector perpendicular to two known vectors. Remark 9 Using the definition, it is easy to verify that i × j = k j × k = i k × i = j and j × i = −k k × j = −i i × k = −j Remark 10 From property 3 of theorem 7, it follows that two non-zero vectors are parallel if and only if their cross product is 0.

The cross product satisfies more properties which we will not prove because they are very tedious. Theorem 11 Let u, v , and w be three vectors and a be a scalar. The following is true: 1. u × v = −v × u (this tells us that the cross product is not commutative. 2. (a u) × v = a ( u × v ) = u × (av) 3. u × (v + w) = u × v + u×w 4. ( u + v) × w = u×w + v × w

1.2.1

Area of a Parallelogram

Consider a parallelogram whose sides are given by the vectors u and v as shown in the figure below. Remembering that the area of a parallelogram is the length of its base times its height, we see that the area A of this parallelogram is A = u v sin θ = u × v

→ − u ×→ v= Area of the paralelogram is − − − → → u v sin θ

Example 12 Find the area of the parallelogram shown in the figure below.

Find the area

1.3

Triple Products

Definition 13 Given three non-zero vectors u, v , and w, the product u · (v × w) is called the scalar triple product of the vectors u, v, and w. Proposition 14 The volume of the parallelepiped determined by the vectors u, v, and w as shown below is the magnitude of their scalar triple product | u · (v × w)|.

− − → Parallelepiped determined by → u, → v and − w

Proof. The volume V of a parallelepiped is given by V = area of the base times height

Suppose the base of the parallelepiped is determined by v and w. Let θ be the angle u makes with the direction perpendicular to the base. Then the height of the parallelepiped is | u cos θ|. The area of the base is v × w. Therefore, V

= |v × w u cos θ| = | u · (v × w)|

Corollary 15 Three non-zero vectors u, v, and w are coplanar (on the same plane) if u · (v × w) = 0. Remark 16 If instead of thinking of the parallelepiped as having its base determined by v and w, we had thought of it as having its base determined by u and v, then we would have found that its volume was |w · ( u × v)|. But since we are talking about the same parallelepiped, the two formulas for the volume must be the same, so we have: u · (v × w) =w · ( u × v )

(1)

Remark 17 The scalar triple product of three non-zero vectors u, v , and w can be computed by calculating the determinant

u

x uy uz

u · (v × w) = vx vy vz

wx wy wz

(2)

1.4

Summary

The cross product is a very important quantity in mathematics. It can be used for: 1. Find a vector perpendicular to two non-zero vectors (often used in computer graphics). 2. Find the area of a parallelogram. 3. Find the volume of a parallelepiped. 4. Determine if two non-zero vectors are parallel. 5. Determine if three non-zero vectors are coplanar. 6. Many applications in physics which we will not discuss here.

1.5

Vectors and Maple

To handle vectors using Maple 9.5, one must first load the LinearAlgebra package with the command with(LinearAlgebra); Once this package is loaded, the following operations can be performed: • Defining a vector: This is done using the construct , , . ⎤For example, to define the vector A to be ⎡ 1 ⎢ ⎥ ⎣ 3 ⎦, use −4 A := 1, 3, −4 ; • Adding two vectors: Use the usual addition symbol as in A+B

• Scalar Multiplication: Use the usual multiplication symbol as in 2∗A • Subtracting two vectors: Use the usual subtraction symbol as in A−B • Finding the norm of a vector: The norm we defined in this class is called the 2-norm in more advanced mathematics classes because we take the square root of the sum of the squares of the coordinates. To do this with Maple, use Norm(A, 2); where A is a vector. • Dot product: Given two vectors A and B, their dot product can be found using DotProduct(A,B);

or the shortcut A.B; • Cross product: Given two vectors A and B, their cross product can be found using CrossProduct(A,B); or the shortcut A &x B; There must be spaces between A and & as well as between x and B. • Plotting vectors: To plot vectors, one must first load the plots package with the command with(plots); To plot the vector A, one would then use arrow(A, shape=arrow);

The shape parameter is optional. To plot two or more vectors, one must list the vectors inside square brackets. The command is: arrow([A,B],shape=arrow); To find all the parameters of the arrow command, use the help facility of Maple.

1.6

Problems

Odd numbers 1-27 except 5 and 19. Also, do #2 and 28 on pages 664, 665.