Lesson 3: The Cross Product

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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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Lesson 3: The Cross Product

  1. 1. Section 9.4 Cross Products and Planes Math 21a February 8, 2008 Announcements Homework for Monday 2/11: Section 9.4. Exercises 4, 6, 8, 10, 19, 22, 30; pp. 664–665. Section 9.5. Exercise 1*; pp. 673–675.
  2. 2. Outline Torque and the Cross Product Properties of the Cross Product On a basis In General By components Other applications Area Volume
  3. 3. Torque When force is applied to a lever fixed to a point, some of the force goes towards rotation while the rest goes towards stretching the lever. τ r |F | sin θ θ F
  4. 4. Torque When force is applied to a lever fixed to a point, some of the force goes towards rotation while the rest goes towards stretching the lever. τ r |F | sin θ θ F The magnitude of the torque is also proportional to the length of the lever, and has a direction depending on which direction the lever pivots.
  5. 5. Example z gq o CD CD c) o o qgl rA o o a o Fl a A bicycle pedal is pushed by a foot with a 60 N force. The crank K a oo TI v') 0q C) rn o o I quot;j arm is 180 mm long. Find the magnitude of the torque about P. (h CD CD o CD o c) (D CF oc w (D o o I d 0 a o 15 (h a {
  6. 6. Example z gq o CD CD c) o o qgl rA o o a o Fl a A bicycle pedal is pushed by a foot with a 60 N force. The crank K a oo TI v') 0q C) rn o o I quot;j arm is 180 mm long. Find the magnitude of the torque about P. (h CD CD o CD o c) (D CF oc w (D o o I d 0 a o 15 (h a Solution { |τ | = |r| |F| |sin θ| = (0.18 m)(60 N) sin(80◦ ) ≈ 10.6359 N m
  7. 7. In General Definition Given vectors a and b in space, the cross product of a and b is the vector a × b = |a| |b| (sin θ) n, where n is a vector perpendicular to a and b such that (a, b, n) is a right-handed set of three vectors.
  8. 8. 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) 4. (a · b) × c 2. a × (b · c) 5. (a · b) × (c · d) 3. a × (b × c) 6. (a × b) · (c × d)
  9. 9. Outline Torque and the Cross Product Properties of the Cross Product On a basis In General By components Other applications Area Volume
  10. 10. Cross products of the standard basis vectors Fill in the table: × i j k i j k
  11. 11. Cross products of the standard basis vectors Fill in the table: × i j k i 0 k −j j −k 0 i k j −i 0 Is the cross product commutative? Is the cross product associative?
  12. 12. Cross products of the standard basis vectors Fill in the table: × i j k i 0 k −j j −k 0 i k j −i 0 Is the cross product commutative? Is the cross product associative?
  13. 13. Cross products of the standard basis vectors Fill in the table: × i j k i 0 k −j j −k 0 i k j −i 0 Is the cross product commutative? No i × j = k = −j × i Is the cross product associative?
  14. 14. Cross products of the standard basis vectors Fill in the table: × i j k i 0 k −j j −k 0 i k j −i 0 Is the cross product commutative? No i × j = k = −j × i Is the cross product associative?
  15. 15. Cross products of the standard basis vectors Fill in the table: × i j k i 0 k −j j −k 0 i k j −i 0 Is the cross product commutative? No i × j = k = −j × i Is the cross product associative? No i × (i × j) = i × k = −j (i × i) × j = 0
  16. 16. 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
  17. 17. Cross product by components Question If a = a1 , a2 , a3 = a1 i + a2 j + a3 k b = b1 , b2 , b3 = b1 i + b2 j + b3 k Find a × b.
  18. 18. Cross product by components Question If a = a1 , a2 , a3 = a1 i + a2 j + a3 k b = b1 , b2 , b3 = b1 i + b2 j + b3 k Find a × b. Answer a × b = (a2 b3 − b2 a3 )i + (a3 b1 − b3 a1 )j + (a1 b2 − b1 a2 )k = a2 b3 − b2 a3 , a3 b1 − b3 a1 , a1 b2 − b1 a2
  19. 19. Determinant formula This is only to help you remember, in case you’ve seen determinants of 3 × 3 matrices: i j k a a a a a a a1 a2 a3 = i 2 3 − j 1 3 + k 1 2 b2 b3 b1 b3 b1 b2 b1 b2 b3 = (a2 b3 − b2 a3 )i − (b3 a1 − a3 b1 )j + (a1 b2 − b1 a2 )k =a×b
  20. 20. 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, t 2 , t 3 and b = 1, 2t, 3t 2
  21. 21. 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, t 2 , t 3 and b = 1, 2t, 3t 2 Solution 1. 2, −1, 3 2. 2i + 13j − 8k 3. t 4 , −2t 3 , t 2
  22. 22. Outline Torque and the Cross Product Properties of the Cross Product On a basis In General By components Other applications Area Volume
  23. 23. Area The magnitude of the cross product a × b is the area of the parallelogram with sides a and b. b |b| sin θ a
  24. 24. Volume To find the volume of a paralleliped with sides a, b, c: c b a
  25. 25. Volume To find the volume of a paralleliped with sides a, b, c: c b a We get V = |a · (b × c)|
  26. 26. More determinants a · (b × c) = a1 , a2 , a3 · b2 c3 − c2 b3 , b3 c1 − c3 b1 , b1 c2 − c1 b2 = a1 (b2 c3 − c2 b3 ) + a2 (b3 c1 − c3 b1 ) + a3 (b1 c2 − c1 b2 ) a1 a2 a3 = b1 b2 b3 c1 c2 c3
  27. 27. Example Example Find the volume of the parallelepiped determined by a = 6i + 3j − k b = j + 2k c = 4i − 2j + 5k.
  28. 28. Example Example Find the volume of the parallelepiped determined by a = 6i + 3j − k b = j + 2k c = 4i − 2j + 5k. Solution The volume is 6 3 −1 0 1 2 = 6(5 + 4) − 3(0 − 8) − 1(−4) = 54 + 24 + 4 = 82 4 −2 5
  29. 29. Cross product jokes What do you get when you cross a lion with a tiger?
  30. 30. Cross product jokes What do you get when you cross a lion with a tiger? What do you get when you cross a lion with a mountain climber?
  31. 31. Cross product jokes What do you get when you cross a lion with a tiger? What do you get when you cross a lion with a mountain climber? What do you get when you cross a mosquito with a fish monger?
  32. 32. Cross product jokes What do you get when you cross a lion with a tiger? What do you get when you cross a lion with a mountain climber? What do you get when you cross a mosquito with a fish monger? What do you get when you cross an elephant with a banana?
  33. 33. Cross product jokes What do you get when you cross a lion with a tiger? What do you get when you cross a lion with a mountain climber? What do you get when you cross a mosquito with a fish monger? What do you get when you cross an elephant with a banana? What do you get when you cross a mathematician with a movie star?

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