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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?