Calculus 2nd Edition Rogawski

694 C H A P T E R 13 VECTOR GEOMETRY
FIGURE 1 The spiral paths of charged
particles in a bubble chamber in the
presence of a magnetic field are described
using cross products.
FIGURE 2 The Van Allen radiation belts,
located thousands of miles above the
earth’s surface, are made up of streams of
protons and electrons that oscillate back
and forth in helical paths between two
“magnetic mirrors” set up by the earth’s
magnetic field. This helical motion is
explained by the “cross-product” nature
of magnetic forces.
The theory of matrices and determinants is
part of linear algebra, a subject of great
importance throughout mathematics. In
this section, we discuss just a few basic
definitions and facts needed for our
treatment of multivariable calculus.
13.4 The Cross Product
This section introduces the cross product v × w of two vectors v and w. The cross product
(sometimes called the vector product) is used in physics and engineering to describe
quantities involving rotation, such as torque and angular momentum. In electromagnetic
theory, magnetic forces are described using cross products (Figures 1 and 2).
Unlike the dot product v · w (which is a scalar), the cross product v × w is again a
vector. It is defined using determinants, which we now define in the 2 × 2 and 3 × 3 cases.
A2× 2 determinant is a number formed from an array of numbers with two rows and two
columns (called a matrix) according to the formula
∣ a
c
b
d ∣ = ad − bc 1
Note that the determinant is the difference of the diagonal products. For example,
∣ 3 2
∣∣∣ ∣ 1
2
4 ∣ = 3 2
1
2
4 ∣ −
∣ 3 2
1
2
4 ∣ = 3 · 4 − 2 · 1
2 = 11
The determinant of a 3 × 3 matrix is defined by the formula
∣
∣
a 1 b 1 c 1 ∣∣∣∣∣ ∣ ∣ ∣∣∣ b
a 2 b 2 c 2 = a 2 c 2 ∣∣∣
1
b
a 3 b 3 c 3 c 3 3
(1, 1)-minor
∣ ∣ ∣ ∣ ∣∣∣ a
− b 2 c 2 ∣∣∣ ∣∣∣ a
1 + c 2 b 2 ∣∣∣
c 1 3 b 3
a 3
(1, 2)-minor
a 3
(1, 3)-minor
This formula expresses the 3 × 3 determinant in terms of 2 × 2 determinants called
minors. The minors are obtained by crossing out the first row and one of the three columns
of the 3 × 3 matrix. For example, the minor labeled (1, 2) above is obtained as follows:
∣ a 1 b 1 c 1 ∣∣∣∣∣
a 2 b 2 c 2 to obtain the (1, 2)-minor
∣ a ∣
2 c 2 ∣∣∣
∣
a
a 3 b 3 c 3 c 3 3
Cross out row 1 and column 2
EXAMPLE 1 A 3 × 3 Determinant Calculate
Solution
∣
2 4 3
0 1 −7
−1 5 3
∣
2 4 3
0 1 −7
−1 5 3
∣ .
(1,2)-minor
∣ ∣ ∣ ∣∣∣ ∣ = 2 1 −7
∣∣∣ 5 3 ∣ − 4 0 −7
∣∣∣ −1 3 ∣ + 3 0 1
−1 5 ∣
= 2(38) − 4(−7) + 3(1) = 107
2
Later in this section we will see how determinants are related to area and volume.
First, we introduce the cross product, which is defined as a “symbolic” determinant whose
first row has the vector entries i, j, k.
CAUTION Note in Eq. (3) that the middle
term comes with a minus sign.
DEFINITION The Cross Product The cross product of vectors v = ⟨a 1 ,b 1 ,c 1 ⟩ and
w = ⟨a 2 ,b 2 ,c 2 ⟩ is the vector
∣ i j k ∣∣∣∣∣ v × w =
a 1 b 1 c 1 =
∣ b ∣ 1 c 1 ∣∣∣ i −
∣
b
a 2 b 2 c 2 c 2
∣ a ∣ 1 c 1 ∣∣∣ j +
a 2 c 2
∣ a 1
a 2 2
∣
b 1 ∣∣∣
k
b 2
3