Pre-Calc Vectors ~1~ NJCTL.org

Transcription

1 Intro to Vectors Class Work Draw vectors to represent the scenarios. 1. A plane flies east at 300 mph. 2. A ship sails northwest at 20 knots. 3. A river flows south at 4 mph. Draw the following vector. Show the component forces for the given vectors. 4. a = (6, 8) 5. C = ( 3,7) 6. u joins (2,3) to (4, 9) 7. v joins (4, 9) to (2,3) Referring to Questions 4-7, find the following 8. a 9. C 10. u 11. v Referring to Questions 4-7, find the direction of the vector 12. a 13. C 14. u 15. v Intro to Vectors Homework Draw vectors to represent the scenarios. 16. A plane flies west at 200 mph. 17. A ship sails northeast at 10 knots. 18. A river flows north at 3 mph. Draw the following vector. Show the component forces for the given vectors. 19. a = ( 3, 4) 20. C = (5, 12) 21. u joins (1,4) to (8,6) 22. v joins ( 2,3) to (3, 2) Referring to Questions 16-19, find the following 23. a 24. C 25. u 26. v Referring to Questions 16-19, find the direction of the vector 27. a 28. C 29. u 30. v Pre-Calc Vectors ~1~ NJCTL.org

2 Converting Between Rectangular and Polar Forms Class Work Convert the following Polar coordinates to rectangular. 31. (4, 45 ) 32. (3, 60 ) 33. (10, 110 ) 34. (7, π 5 ) 35. (12, 4π 7 ) Convert the following rectangular coordinates to polar. 36. (5, 8) 37. (4, -9) 38. (-3,6) 39. (-4,-2) 40. (0, -3) Converting Between Rectangular and Polar Forms Homework Convert the following Polar coordinates to rectangular. 41. (5, 135 ) 42. (6, 30 ) 43. (12,170 ) 44. (5, 3π 5 ) 45. ( 12, 3π 7 ) Convert the following rectangular coordinates to polar. 46. (4, 2) 47. (3, -9) 48. (12, -12) 49. (-7, 8) 50. (-4, -5) Pre-Calc Vectors ~2~ NJCTL.org

3 Scalar Multiplication Class Work Given u = (4,2) and v ( 2,3), find the following and draw the transformation u 52. 3v u 54. v 2 Scalar Multiplication Homework Given u = (5, 4) and v ( 3, 2), find the following and draw the transformation u 56. 3v u 58. v 2 Pre-Calc Vectors ~3~ NJCTL.org

4 Vector Addition Class Work Use the vectors to draw the expression. Draw the resultant vector. 59. E + F 60. D + B 61. A + C 62. 2C 63. 2B + F Find the resultant vector. State the magnitude and direction of each resultant vector. 64. u = (4,2) and v = (3,2), find u + v 65. u = ( 5,6) and v = ( 3,2), find u + v 66. A tug-of-war contest is made up of 2-member teams. Consider the rope to be on the x-axis with the flag at the origin. Team A s members pull (3,2) and (4,-1). Team B s members pull (-4,0) and (-3,0). What is the magnitude and direction of each team s actions? What is the movement of the flag? Pre-Calc Vectors ~4~ NJCTL.org

5 Vector Addition Homework Use the vectors to draw the expression. Draw the resultant vector. 67. B + F 68. E + B 69. D + C 70. 3D 71. E + 2F Find the resultant vector. State the magnitude and direction of each resultant vector. 72. u = (5,1) and v = ( 5,2), find u + v 73. u = ( 5,3) and v = ( 3,2), find u + v 74. A tug-of-war contest is made up of 2-member teams. Consider the rope to be on the x-axis with the flag at the origin. Team A s members pull (4,3) and (4,-2). Team B s members pull (-5,0) and (-4,0). What is the magnitude and direction of each team s actions? What is the movement of the flag? Pre-Calc Vectors ~5~ NJCTL.org

6 Vector Subtraction Class Work Use the vectors to draw the expression. Draw the resultant vector. 75. B F 76. E B 77. D C 78. 3D A 79. E 2F + B Find the resultant vector. State the magnitude and direction of each resultant vector. 80. u = (5,1) and v = ( 5,2), find u v 81. u = ( 5,3) and v = ( 3,2), find u v 82. u = ( 3,1) and v = ( 4, 2), find 2u 3v Pre-Calc Vectors ~6~ NJCTL.org

7 Vector Subtraction Homework Use the vectors to draw the expression. Draw the resultant vector. 83. A C 84. D F 85. C B 86. 3C F 87. D 3A + C Find the resultant vector. State the magnitude and direction of each resultant vector. 88. u = (2, 3) and v = ( 4,2), find u v 89. u = ( 4,6) and v = ( 7, 2), find u v 90. u = ( 5,4) and v = ( 8,0), find 3u 2v Pre-Calc Vectors ~7~ NJCTL.org

8 Vector Equations of a Line Class Work Write the vector equation of the line and the parametric equation for the line: 91. through (7, 4) and parallel to v = (1, 2). 92. through (-3, 5) and parallel to v = (4,3). 93. through (11, 0) and parallel to v = ( 7,0). 94. through (-5, -8) and parallel to v = (0,8). 95. through (6, -1) and parallel to v = ( 9, 10). 96. through (2, 8) and (5, 9) x = 2 + 4t 97. through (0, 3) and (7,0) 98. Write the vector equation for { y = 3 7t Vector Equations of a Line Homework Write the vector equation of the line and the parametric equation for the line: 99. through (3, 9) and parallel to v = (2, 5) through (-11, 13) and parallel to v = (6,10) through (2, 14) and parallel to v = ( 11,2) through (-4, -9) and parallel to v = (3,18) through (1, -3) and parallel to v = ( 12, 11) through (5, 7) and (-4, 3) 105. through (1, 7) and (-4,7) 106. Write the vector equation for { x = 2 + 5t y = 3t Pre-Calc Vectors ~8~ NJCTL.org

9 Dot Product Class Work Find the dot product of the vectors. State whether they are perpendicular or form an obtuse or acute angle a = (2,4) and b = (3,5) 108. a = (3, 2) and b = (4,6) 109. a = ( 2,1) and b = (4, 2) 110. a = ( 5,8) and b = (10,6) 111. a = (8, 4) and b = (3,6) 112. a = ( 4,6) and b = (9, 6) Dot Product Homework Find the dot product of the vectors. State whether they are perpendicular or form an obtuse or acute angle a = (3,6) and b = (2, 9) 114. a = (8,4) and b = (3, 6) 115. a = (10,8) and b = (4,5) 116. a = (3,4) and b = (9,12) 117. a = (0,9) and b = (7,5) 118. a = ( 2,8) and b = (4,1) Pre-Calc Vectors ~9~ NJCTL.org

10 Angles Between Vectors Class Work Find the angle between the two given vectors a = (2,4) and b = (7,1) 120. c = ( 1,4) and d = (8,2) 121. d = ( 3,0) and e = (3, 1) 122. f = (4, 3)and g = ( 1, 2) 123. h = (2, 6)and i = ( 1,3) 124. j = ( 1,4)and k = (2,0) Angles Between Vectors Homework Find the angle between the two given vectors a = (3,5) and b = (7,2) 126. c = ( 2,4) and d = (8,1) 127. d = ( 5, 1) and e = (2, 1) 128. f = (5, 3)and g = ( 4, 7) 129. h = (4, 8)and i = ( 2,3) 130. j = ( 1,6)and k = ( 1,0) Pre-Calc Vectors ~10~ NJCTL.org

11 3-Dimensional Space Class Work 131. What is the distance between (1,2,3) and (4,5,6)? 132. What is the distance between (-4,0,-7) and (3,-2,-9)? 133. How far is (5,3,-4) from the origin? 134. What is the length of a diagonal of a box with sides 4x4x8? 135. What is radius and the center of the sphere with equation: (x-2) 2 + y 2 + (z-4) 2 = 36? 136. What is radius and the center of the sphere with equation: x 2 + 6x+ y 2 + (z-7) 2 = 16? 137. What is radius and the center of the sphere with equation: x 2-6x+ y 2 +8y+ z 2-10z= -1? Pre-Calc Vectors ~11~ NJCTL.org

12 3-Dimensional Space Homework 138. What is the distance between (10,6,2) and (3,5,7)? 139. What is the distance between (-2,1,-5) and (4,-6,-3)? 140. How far is (-5,-3,-4) from the origin? 141. What is the length of a diagonal of a box with sides 5x7x6? 142. What is radius and the center of the sphere with equation: (x+3) 2 + y 2 + (z+5) 2 = 64? 143. What is radius and the center of the sphere with equation: x x+ y 2 + (z-8) 2 = -16? 144. What is radius and the center of the sphere with equation: x 2-10x+ y 2 +4y+ z 2 +20z=15? Pre-Calc Vectors ~12~ NJCTL.org

13 Vectors, Lines, and Planes Class Work Given u = (1,2, 3) and v = ( 4,5, 6) compute the following u + v 146. u v u + 2v 148. u v 149. u 150. v 151. the angle between u and v 152. Write the vector equation of the line through u and v 153. Write the equation from #152 in parametric form u v Vectors, Lines, and Planes Homework Given u = (5,3, 4) and v = ( 2,6,0) compute the following u + v 156. u v u + 2v 158. u v 159. u 160. v 161. the angle between u and v 162. Write the vector equation of the line through u and v 163. Write the equation from #162 in parametric form u v Pre-Calc Vectors ~13~ NJCTL.org

14 Multiple Choice Unit Review 1. A vector has component forces of Ax= 5.2 and Ay= -4.7, what is A? a b c d u = (1,4), find 5 4 u. a. ( 5 4, 5) b. ( 9, 21 ) 4 4 c. (9,4) 25 d a = (4, 6) and b = ( 2,5), find b + a. a. (6,11) b. (2,-11) c. (6,-1) d. (2,-1) 4. a = (4, 6) and b = ( 2,5), find 2b a. a. (8,-12) b. (2,-1) c. (-8,16) d. (-10,-17) 5. a = (4, 6) and b = ( 2,5), find b a. a. 16 b. -16 c. -38 d What is the slope of the line with vector equation (x-3,y+5)=t(2,6) a. 2 b. 3 c. 5 d An example of perpendicular vectors is a. u = (4,5) and v = ( 2,3) b. u = (2, 6) and v = ( 9,3) c. u = ( 3,4) and v = ( 8,6) d. u = ( 4,6) and v = ( 9, 6) 8. The angle between a = (4, 6) and b = ( 2,5) is a..979 b c d Pre-Calc Vectors ~14~ NJCTL.org

15 9. What is the distance between (4,-2,-5) and (-1, 7,-6)? a b c d Which of the following points is 12 units from the origin? a. (3,4,5) b. (0,12,1) c. (-5,12,0) d. (-4, 8, 8) 11. What is the radius of x 2 +8x + y 2 12y + z 2 + 2z - 9=16? a. 4 b. 5 c. 31 d Given u = (5, 3, 7) and v = ( 2, 1, 8), find u v. a. -59 b. -46 c. 49 d Given u = (5, 3, 7) and v = ( 2, 1, 8), find v a b c d Given u = (5, 3, 7) and v = ( 2, 1, 8), find u v a. (-17, 54, 11) b. (-31, -26, -1) c. (-17,-54, -1) d. (-10, 3, 56) Extended Response 1. A plane flies northeast at 300 miles per hour. a. Draw a vector representation of the plane. Show the component forces. b. The wind is blowing south at 50 miles per hour, what is the result on the planes component forces? c. Where will the plane be in 5 hours relative to its starting position? Pre-Calc Vectors ~15~ NJCTL.org

16 2. A box is being slid across the floor by a person pulling a rope with component force of Fx=6 and Fy=3. a. Another person pulls a second rope with Fx= -4 and Fy=3. Draw a vector diagram to model this situation. b. Where does the box end up? c. If the second person had wanted the box to slide due north, what component forces should they have applied? Explain. 3. Three people are holding the lines of a balloon during a parade. a = (5,7, 4), b = (3, 18, 2), and c = ( 2, 3, 1). a. What is direction of the balloon? b. What is the angle of the lines between person A and person B? c. What vector would represent the effects of helium on the balloon? 4. Line m passes through (-6, 4, -8) and (2, 7, -9). a. Write the vector equation of line m. b. Write the parametric equation of line. c. Find a point, other than the ones given, that lies on the line. Pre-Calc Vectors ~16~ NJCTL.org