MAT 1236Calculus III

Section 12.4

The Cross Product

http://myhome.spu.edu/lauw

HW…

WebAssign 12.4(18 problems, 98 min.)

Read 12.5 (Seriously!): The first not-too-easy section in Calculus

Preview

Define a new operation on vectors: The Cross Product

Unlike the dot product, the cross product of two vectors is a vector.

Properties of the cross product.

Classwork of the last section...

We did not have time to work on the last classwork....

???

v

u

The Right Hand Rule

FBI

We are Interested in …

Given 2 vectors, they “span” a plane Find a vector perpendicular to this plane

The Cross Product

If and , the cross product of a and b is the vector

1 2 3, ,a a a a 1 2 3, ,b b b b

2 3 3 2 3 1 1 3

1 2 3 1 2 3

2 3 3 2 3 1

1 2 2

1 1 2 2 1

1

3

, , , ,

, ,

a b a a a b b b

a b a b a b a b a

a b a b i a b a b j a b a b

b b

k

a

The Cross Product

The formula is traditionally memorized by using (formal) determinant expansions

2 3 3 2 3 1 1 3

1 2 3 1 2 3

2 3 3 2 3 1

1 2 2

1 1 2 2 1

1

3

, , , ,

, ,

a b a a a b b b

a b a b a b a b a

a b a b i a b a b j a b a b

b b

k

a

2x2 Determinant Expansions

a bad bc

c d

3x3 Determinant Expansions

a b ce f d f d e

d e f a b ch i g i g h

g h i

3x3 Determinant Expansions

a b ce f d f d e

d e f a b ch i g i g h

g h i

3x3 Determinant Expansions

a b ce f d f d e

d e f a b ch i g i g h

g h i

3x3 Determinant Expansions

a b ce f d f d e

d e f a b ch i g i g h

g h i

The Cross Product

The formula is traditionally memorized by using (formal) determinant expansions

1 2 3

1 2 3

i j k

a b a a a

b b b

2 3 3 2 3 1

1 2 3 1 2 3

1 3 1 2 2 1

, , , ,

a b a

a b a a a

b i a b a b j

b b b

a b a b k

Example 1

2 3 , a i j k b i k

i j k

a b

Expectations

You are expected to use the above standard procedure to find the cross product.

You are expected to show all the steps. Keep in mind, good practices are key to minimize the chance of making mistakes.

Property A

0a a

1 2 3

1 2 3

i j k

a a a a a

a a a

Property B

a b a

a b b

is orthogonal to both and a b a b

Property B

a b a

a b b

is orthogonal to both and a b a b

In addition, the cross product obeys the Right Hand Rule.

Property B (Why?) is orthogonal to both and a b a b

2 3 1 3 1 21 2 3

2 3 1 3 1 21 2 3

i j ka a a a a a

a b a a a i j kb b b b b b

b b b

Example 1 (Verify Property B) 3, , 33 32a i j k b i k a b i j k

a b a

a b b

Property C

sin , 0a b a b

Property C (Why?)

sin , 0a b a b

In Particular

In Particular

is in the same direction of k and

i j

i j

Property D

Two nonzero vectors and are parallel if and only if 0a b

Property D (Why?)

Two nonzero vectors and are parallel if and only if 0a b

0a b

Property E

The length of the cross product axb is equal to the area of the parallelogram

determined by a and b.

sinA a b a b

Example 2

Find a vector perpendicular to the plane that passes through the points

P(6,0,0) , Q(1,1,1), R(0,0,2)

Example 3

Find the area of the triangle with vertices

P(6,0,0) , Q(1,1,1), R(0,0,2)

Other Properties

Reference only

Default

Right Hand Rule