The Cross Product

The Cross ProductThe Cross ProductThe Cross ProductThe Cross Product

Third Type of Multiplying VectorsThird Type of Multiplying Vectors

Cross Products

1 1 1 2 2 2

1 2 2 1 1 2 2 1 1 2 2 1

If v and

are two vectors in space, the cross product

v is defined as the vector

v ( ) ( ) ( )

a i b j c k w a i b j c k

w

w b c b c i a c a c j a b a b k

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Determinants• It is much easier to do this using

determinants because we do not have to memorize a formula.

• Determinants were used last year when doing matrices

• Remember that you multiply each number across and subtract their

products

Finding Cross Products Using Equation

2 3

3 2 Find v

( 3 1 2 1 (2 ( 1) 3 1) (2 ( 2) ( 3 3)

(3 2) ( 2 3) ( 4 9)

5 5 5

v i j k

w i j k w

i j k

i j k

i j k

��

Evaluating a Determinant

2 32 2 1 3 4 3 1

1 2

6 5(6 1) 5 2 6 10 4

2 1

Evaluating Determinants

1 4 2 4 2 12 1 4

3 1 1 1 1 31 3 1

(1 1 3 4) 2 1 1 4 2 3 1 1

11 2 5

A B C

A B C

A B C

A B C

Using Determinants to Find Cross Products

• This concept can help us find cross products.

• Ignore the numbers included in the column under the vector that will be inserted when setting up the determinant.


• Find v x w given• v = i + j• w = 2i + j + k


1 0 1 0 1 11 1 0

1 1 2 1 2 12 1 1

(1 0) (1 0) (1 2)

i j k

i j k

i j k

i j k


• If v = 2i + 3j + 5k and w = i + 2j + 3k,

• find • (a) v x w• (b) w x v• (c) v x v


( )

2 3 5

1 2 3

3 5 2 5 2 3

2 3 1 3 1 2

(9 10) (6 5) (4 3)

a

i j k

v w

i j k

i j k

i j k

��


( )

1 2 3

2 3 5

2 3 1 3 1 2

3 5 2 5 2 3

(10 9) (5 6) 3 4

Notice that this is the same values as but

the signs are opposite. (Neg. is now pos. and vice

versa)

b

i j k

w v

i j k

i j k

i j k

v w

��

��


2 3 5

2 3 5

3 5 2 5 2 3

3 5 2 5 2 3

(15 15) (10 10) (6 6)

0

This leads us to one of the properties of cross

products.

i j k

v v

i j k

i j k

Algebraic Properties of the Cross Product

• If u, v, and w are vectors in space and if is a scalar, then

• u x u = 0• u x v = -(v x u)• (u x v) = (u) x v = u x (v)• u x (v + w) = (u x v) + (u x w)

Examples• Given u = 2i – 3j + k v = -3i + 3j

+ 2k• w = i + j + 3k• Find• (a) (3u) x v• (b) v . (u x w)

Examples(3 ) 3(2 3 ) 6 9 3

(3 ) 6 9 3

3 3 2

9 3 6 3 6 9

3 2 3 2 3 3

( 18 9) (12 ( 9) (18 27)

27 21 9

u i j k i j k

i j k

u v

i j k

i j k

i j k

Examples

2 3 1

1 1 3

3 1 2 1 2 3

1 3 1 3 1 1

( 9 1) (6 1) (2 ( 3))

10 5 5

( 3 ( 10)) (3 ( 5)) (2 5)

30 15 10 25

i j k

u w

i j k

i j k

i j k

v u w

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Geometric Properties of the Cross Product

• Let u and v be vectors in space• u x v is orthogonal to both u and v.• ||u x v|| = ||u|| ||v|| sin where is

the angle between u and v.• ||u x v|| is the area of the

parallelogram having u ≠ 0 and v ≠ 0 as adjacent sides

Geometric Properties of the Cross Product

• u x v = 0 if and only if u and v are parallel.

Finding a Vector Orthogonal to Two Given

Vectors• Find a vector that is orthogonal to • u = 2i – 3j + k and v = i + j + 3k

• According to the preceding slide, u x v is orthogonal to both u and v. So to find the vector just do u x v


Vectors

2 3 1

1 1 3

3 1 2 1 2 3

1 3 1 3 1 1

( 9 1) (6 1) (2 ( 3))

10 5 5

i j k

u v

i j k

i j k

i j k


Vectors • To check to see if the answer is

correct, do a dot product with one of the given vectors. Remember, if the dot product = 0 the vectors are orthogonal


Vectors

(2 10) ( 3 5) (1 5)

0

u u v

Finding the Area of a Parallelogram

• Find the area of the parallelogram whose vertices are P1 = (0, 0, 0),

• P2 = (3,-2, 1), P3 = (-1, 3, -1) and • P4 = (2, 1, 0)

• Two adjacent sides of this parallelogram are u = P1P2 and v = P1P3.

Finding the Area of the Parallelogram

1 2

1 3

1 2 1 3

1 2 1 3

2 2 2

3 0, 2 0,1 0 3 2

( 1 0,3 0, 1 0) 3

3 2 1 2 7

1 3 1

Area of a Parallelogram =

1 2 7 1 4 49 54 3 6

PP i j k

PP i j k

i j k

PP PP i j k

PP PP

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Your Turn• Try to do page 653 problems 1 –

47 odd.

The Cross Product

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