Download ppt - Section 11.4 The Cross Product Calculus III September 22, 2009 Berkley High School

Section 11.4The Cross Product

Calculus IIISeptember 22, 2009Berkley High School

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Operations on Vectors(So Far) Scalar Multiplication

Vector Addition

Vector Subtraction

Dot Product

c a

a b

a b a b

cosa b a b

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New Operation: Cross Product

something with determinantsa b

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Definition of Determinant

deta b a b

ad bcc d c d

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Definition of Determinant

1 2 32 3 1 3 1 2

1 2 3 1 2 32 3 1 3 1 2

1 2 3

1 2 3 3 2

2 1 3 3 1

3 1 2 2 1

a a ab b b b b b

b b b a a ac c c c c c

c c c

a b c b c

a b c b c

a b c b c

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Definition of Cross Product

1 2 3 1 2 3

2 3 1 3 1 21 2 3

2 3 1 3 1 21 2 3

2 3 3 2 1 3 3 1 1 2 2 1

Let , , , , ,a a a a b b b b

i j ka a a a a a

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

b b b

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

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Example

1,1,1 2,3,4

1 1 1

2 3 4

1 4 1 3 1 4 1 2 1 3 1 2

1 2 1 1, 2,1

i j k

i j k

i j k

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Example

1,0,0 0,1,0

1 0 0

0 1 0

0 0 0 1 1 0 0 0 1 1 0 0

0 0 1

i j k

i j k

i j k k

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Properties: Direction & Magnitude

is orthogonal to both and

and uses the right hand rule.

sin

1,0,0 0,1,0 0,0,1

a b a b

a b a b

i j k

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Properties: Direction & Magnitude

2 3 3 2 1 3 3 1 1 2 2 1 1 2 3

1 2 3 3 2 2 1 3 3 1 3 1 2 2 1

1 2 3 1 3 2 2 1 3 2 3 1 3 1 2 3 2 1

Show is orthogonal to both and

0?

, , 0?

0?

0?

0 0

Similarily, it

a b a b

a b a

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

a a b a b a a b a b a a b a b

a a b a a b a a b a a b a a b a a b

can be shown 0a b b

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Properties: Area of Parallelogram

sina b a b

a

b

sinb

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Properties

0

if then 0

a a

a b b a

a b a b

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Assignment

Section 11.4, 1-35, odd x25 But wait, there’s more…

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Volume of Parallelepiped

b

c

a

Volume Bh

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b

c

a

Volume b c h

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b

c

a

h

Volume b c h

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b

c

a

h

cos cosh

h aa

Volume b c h

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b

c

a

h

b c

cos cosh

h aa

Volume b c h

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b

c

a

h

cos cosh

h aa

b c

Volume cosb c a

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b

c

a

h

b c

Volume b c a

Triple Scalar Product

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Example

1 2 3 1 2 3 1 2 3

1 2 3

1 2 3

1 2 3

1 2 3 3 2 2 1 3 3 1 3 1 2 2 1

Property

Let , , , , , , , ,

The triple scalar product

a a a a b b b b c c c c

a a a

a b c b b b

c c c

a b c b c a b c b c a b c b c

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Example

Find the volume of a parallelepiped defined by

1,1, 1 , 1, 1,1 , 1,1,1

1 1 1

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1

2 2 0 4

Volume 4 4

a b c

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Are two vectors coplanar?

Any two non-zero vectors define a plane. Are three vectors coplanar?

The vectors are coplanar iff the parallelepiped formed by the three vectors has Volume=0

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Example

Are 1,4,-7 , 2, 1,4 , 0, 9,18 coplanar?

1 4 7

2 1 4

0 9 18

1 18 36 4 36 0 7 18 0

1 18 4 36 7 18

18 144 126 0

The three vectors are coplanar

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Are three points coplanar?

Any three points define a plane. Are four points coplanar?

From the four points, three vectors can be formed with a common tail. If the vectors are coplanar, then the points are coplanar.

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Assignment

11.4, 41-47 odd