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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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Volume of Parallelepiped
b
c
a
Volume b c h
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Volume of Parallelepiped
b
c
a
h
Volume b c h
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Volume of Parallelepiped
b
c
a
h
cos cosh
h aa
Volume b c h
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Volume of Parallelepiped
b
c
a
h
b c
cos cosh
h aa
Volume b c h
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Volume of Parallelepiped
b
c
a
h
cos cosh
h aa
b c
Volume cosb c a
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Volume of Parallelepiped
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
