Section 9.4: The Cross Product

Practice HW from Stewart Textbook (not to hand in)

p. 664 # 1, 7-17

Cross Product of Two Vectors

The cross product of two vectors produces a vector (unlike the dot product which produces s scalar) that has important properties. Before defining the cross product, we first give a method for computing a determinant.

Definition: The determinant of a matrix,

denoted by , is defined to be the

scalar

22

dc

ba

bcaddc

ba

Example 1: Compute

Solution:

54

32

Definition: If a = i + j + k and b = i + j + k be

vectors in 3D space. The cross product is the

vector

i + j + kba )( 2332 baba )( 1331 baba )( 1221 baba

To calculate the cross product more easily

without having to remember the formula, we

using the following “determinant” form.

2 rowin r left vecto of Components

2 rowin r left vecto of Components

1 rowin rsunit vecto Standard

321

321

b

a

bbb

aaa

kji

ba

We calculate the determinant as

follows: (note the alternation in sign)

33

)()()(

122113312332

21

21

31

31

32

32

321

321

321

321

321

321

321

321

babakbabajbabai

bb

aak

bb

aaj

bb

aai

bbb

aaa

kji

bbb

aaa

kji

bbb

aaa

kji

bbb

aaa

kji

ba

Example 2: Given the vectors

and . Find

a.

kjia 32 kjib 32

ba

b. ab

c. aa

Properties of the Cross Product

Let a, b, and c be vectors, k be a scalar.

1. Note!

2.

3. k (k k b)

4.

5.

)( abba )( abba cabacba )(

)( ba () aba

000 aa0aa

Geometric Properties of the Cross Product

Let a and b be vectors

1. is orthogonal to both a and b.ba

Example 3: Given the vectors

and , show that the cross product

is orthogonal to both a and b.

kjia 32 kjib 32

ba

Note: , , kji jki ikj

x

y

z

2.

3. if and only a = k b, that is, if the vectors a and b are parallel.

4 gives the area having the vectors a and b as its adjacent sides.

sin || || || baba

0ba

|| ba

Example 4: Given the points P(0, -2, 0),

Q(-1, 3, 4), and R(3,0,6).

a. Find a vector orthogonal to the plane through

these points.

b. Find the area of the parallelogram with the

vectors and as its adjacent sides.

c. Find area of the triangle PQR.

Solution: (In the typewritten notes)

PQ

PR