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The cross product is an important operation, taking two three-dimensional vectors and producing a three-dimensional vector. It's not a product in the commutative, associative, sense, but it does produce a vector which is perpendicular to the two crossed vectors and whose length is the area of the parallelogram spanned by the them. The direction is chosen again to follow the right-hand rule.
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Section 9.4Cross Products and Planes
Math 21a
February 8, 2008
Announcements
I Homework for Monday 2/11:I Section 9.4. Exercises 4, 6, 8, 10, 19, 22, 30; pp. 664–665.I Section 9.5. Exercise 1*; pp. 673–675.
Outline
Torque and the Cross Product
Properties of the Cross ProductOn a basisIn GeneralBy components
Other applicationsAreaVolume
Torque
When force is applied to a lever fixed to a point, some of the forcegoes towards rotation while the rest goes towards stretching thelever.
r
F|F | sin θ θ
τ
The magnitude of the torque is also proportional to the length ofthe lever, and has a direction depending on which direction thelever pivots.
Torque
When force is applied to a lever fixed to a point, some of the forcegoes towards rotation while the rest goes towards stretching thelever.
r
F|F | sin θ θ
τ
The magnitude of the torque is also proportional to the length ofthe lever, and has a direction depending on which direction thelever pivots.
Example
A bicycle pedal is pushed by a foot with a 60 N force. The crankarm is 180 mm long. Find the magnitude of the torque about P.
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Solution
|τ | = |r| |F| |sin θ| = (0.18 m)(60 N) sin(80◦) ≈ 10.6359 N m
Example
A bicycle pedal is pushed by a foot with a 60 N force. The crankarm is 180 mm long. Find the magnitude of the torque about P.
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Solution
|τ | = |r| |F| |sin θ| = (0.18 m)(60 N) sin(80◦) ≈ 10.6359 N m
In General
DefinitionGiven vectors a and b in space, the cross product of a and b isthe vector
a× b = |a| |b| (sin θ) n,
where n is a vector perpendicular to a and b such that (a,b,n) isa right-handed set of three vectors.
Example
State whether the following position is meaningful. If not, explain.If so, is the expression a scalar or a vector?
1. a · (b× c)
2. a× (b · c)
3. a× (b× c)
4. (a · b)× c
5. (a · b)× (c · d)
6. (a× b) · (c× d)
Outline
Torque and the Cross Product
Properties of the Cross ProductOn a basisIn GeneralBy components
Other applicationsAreaVolume
Cross products of the standard basis vectors
Fill in the table:× i j ki
0 k − j
j
− k 0 i
k
j − i 0
I Is the cross product commutative?
No
i× j = k = −j× i
I Is the cross product associative?
No
i× (i× j) = i× k = −j
(i× i)× j = 0
Cross products of the standard basis vectors
Fill in the table:× i j ki 0 k − jj − k 0 ik j − i 0
I Is the cross product commutative?
No
i× j = k = −j× i
I Is the cross product associative?
No
i× (i× j) = i× k = −j
(i× i)× j = 0
Cross products of the standard basis vectors
Fill in the table:× i j ki 0 k − jj − k 0 ik j − i 0
I Is the cross product commutative?
No
i× j = k = −j× i
I Is the cross product associative?
No
i× (i× j) = i× k = −j
(i× i)× j = 0
Cross products of the standard basis vectors
Fill in the table:× i j ki 0 k − jj − k 0 ik j − i 0
I Is the cross product commutative? No
i× j = k = −j× i
I Is the cross product associative?
No
i× (i× j) = i× k = −j
(i× i)× j = 0
Cross products of the standard basis vectors
Fill in the table:× i j ki 0 k − jj − k 0 ik j − i 0
I Is the cross product commutative? No
i× j = k = −j× i
I Is the cross product associative?
No
i× (i× j) = i× k = −j
(i× i)× j = 0
Cross products of the standard basis vectors
Fill in the table:× i j ki 0 k − jj − k 0 ik j − i 0
I Is the cross product commutative? No
i× j = k = −j× i
I Is the cross product associative? No
i× (i× j) = i× k = −j
(i× i)× j = 0
Algebraic Properties of the Cross Product
If a, b, and c are vectors and c is a scalar, then
1. a× b = −b× a
2. (ca)× b = c(a× b) = a× (cb)
3. a× (b + c) = a× b + a× c
4. (a + b)× c = a× c + b× c
Cross product by components
QuestionIf
a = 〈a1, a2, a3〉= a1i + a2j + a3k
b = 〈b1, b2, b3〉= b1i + b2j + b3k
Find a× b.
Answer
a× b = (a2b3 − b2a3)i + (a3b1 − b3a1)j + (a1b2 − b1a2)k
= 〈a2b3 − b2a3, a3b1 − b3a1, a1b2 − b1a2〉
Cross product by components
QuestionIf
a = 〈a1, a2, a3〉= a1i + a2j + a3k
b = 〈b1, b2, b3〉= b1i + b2j + b3k
Find a× b.
Answer
a× b = (a2b3 − b2a3)i + (a3b1 − b3a1)j + (a1b2 − b1a2)k
= 〈a2b3 − b2a3, a3b1 − b3a1, a1b2 − b1a2〉
Determinant formula
This is only to help you remember, in case you’ve seendeterminants of 3× 3 matrices:∣∣∣∣∣∣
i j ka1 a2 a3
b1 b2 b3
∣∣∣∣∣∣ = i
∣∣∣∣a2 a3
b2 b3
∣∣∣∣− j
∣∣∣∣a1 a3
b1 b3
∣∣∣∣ + k
∣∣∣∣a1 a2
b1 b2
∣∣∣∣= (a2b3 − b2a3)i− (b3a1 − a3b1)j + (a1b2 − b1a2)k
= a× b
Procedure check
Example
Calculate a× b if
1. a = 〈1, 2, 0〉 and b = 〈0, 3, 1〉2. a = 3i + 2j + 4k and b = i− 2j− 3k
3. a = 〈t, t2, t3〉 and b = 〈1, 2t, 3t2〉
Solution
1. 〈2,−1, 3〉2. 2i + 13j− 8k
3.⟨t4,−2t3, t2
⟩
Procedure check
Example
Calculate a× b if
1. a = 〈1, 2, 0〉 and b = 〈0, 3, 1〉2. a = 3i + 2j + 4k and b = i− 2j− 3k
3. a = 〈t, t2, t3〉 and b = 〈1, 2t, 3t2〉
Solution
1. 〈2,−1, 3〉2. 2i + 13j− 8k
3.⟨t4,−2t3, t2
⟩
Outline
Torque and the Cross Product
Properties of the Cross ProductOn a basisIn GeneralBy components
Other applicationsAreaVolume
Area
The magnitude of the cross product a× b is the area of theparallelogram with sides a and b.
a
b |b| sin θ
Volume
To find the volume of a paralleliped with sides a, b, c:
ab
c
We getV = |a · (b× c)|
Volume
To find the volume of a paralleliped with sides a, b, c:
ab
c
We getV = |a · (b× c)|
More determinants
a · (b× c) = 〈a1, a2, a3〉 · 〈b2c3 − c2b3, b3c1 − c3b1, b1c2 − c1b2〉= a1(b2c3 − c2b3) + a2(b3c1 − c3b1) + a3(b1c2 − c1b2)
=
∣∣∣∣∣∣a1 a2 a3
b1 b2 b3
c1 c2 c3
∣∣∣∣∣∣
Example
Example
Find the volume of the parallelepiped determined by
a = 6i + 3j− k
b = j + 2k
c = 4i− 2j + 5k.
SolutionThe volume is∣∣∣∣∣∣∣∣∣∣∣∣6 3 −10 1 24 −2 5
∣∣∣∣∣∣∣∣∣∣∣∣ = 6(5 + 4)− 3(0− 8)− 1(−4) = 54 + 24 + 4 = 82
Example
Example
Find the volume of the parallelepiped determined by
a = 6i + 3j− k
b = j + 2k
c = 4i− 2j + 5k.
SolutionThe volume is∣∣∣∣∣∣∣∣∣∣∣∣6 3 −10 1 24 −2 5
∣∣∣∣∣∣∣∣∣∣∣∣ = 6(5 + 4)− 3(0− 8)− 1(−4) = 54 + 24 + 4 = 82
Cross product jokes
I What do you get when you cross a lion with a tiger?
I What do you get when you cross a lion with a mountainclimber?
I What do you get when you cross a mosquito with a fishmonger?
I What do you get when you cross an elephant with a banana?
I What do you get when you cross a mathematician with amovie star?
Cross product jokes
I What do you get when you cross a lion with a tiger?
I What do you get when you cross a lion with a mountainclimber?
I What do you get when you cross a mosquito with a fishmonger?
I What do you get when you cross an elephant with a banana?
I What do you get when you cross a mathematician with amovie star?
Cross product jokes
I What do you get when you cross a lion with a tiger?
I What do you get when you cross a lion with a mountainclimber?
I What do you get when you cross a mosquito with a fishmonger?
I What do you get when you cross an elephant with a banana?
I What do you get when you cross a mathematician with amovie star?
Cross product jokes
I What do you get when you cross a lion with a tiger?
I What do you get when you cross a lion with a mountainclimber?
I What do you get when you cross a mosquito with a fishmonger?
I What do you get when you cross an elephant with a banana?
I What do you get when you cross a mathematician with amovie star?
Cross product jokes
I What do you get when you cross a lion with a tiger?
I What do you get when you cross a lion with a mountainclimber?
I What do you get when you cross a mosquito with a fishmonger?
I What do you get when you cross an elephant with a banana?
I What do you get when you cross a mathematician with amovie star?