BC VECTOR PROBLEMS. 13. Find the area of the parallelogram having AB and AC as adjacent sides: A(2,1,3), B(1,4,2), C( 3,2,7) 14.
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1 For problems 9 use: u (,3) v (3, 4) s (, 7). w =. 3u v = 3. t = 4. 7u = u w (,3,5) 5. wt = t (,, 4) 6. Find the measure of the angle between w and t to the nearest degree. 7. Find the unit vector having direction opposite of u. 8. Are the vectors u and s orthogonal? Why or why not? 9. Find wt = 5 7. Find a unit vector to u = (,,3) and v = (5,3,). i k? 3. Find the area of the parallelogram having AB and AC as adjacent sides: A(,,3), B(,4,), C( 3,,7) 4. Solve: 5xyz x3yz 4x 6y 3z 3 5. Graph the ordered triple (not vector!) (3,,5) using the box method. 6. Kathy walks 5 blocks west along a vacant lot and then blocks south to Alice s house. Annie starts at the same point and walks diagonally through the vacant lot coming out at the same point as Alice. If Kathy walked 5 feet west, how far did Annie walk? 7. Write the vector ( 6, 3) as a linear combination of i and j 8. Write t ( 8, 9) as a linear combination of u (, ) and v (6,3) 9. Write t ( 3, ) as a linear combination of u ( 5, ) and v (,) 5 0. Write t (,) 9 as a linear combination of u (,) 3 and v (, 3 ). Find a vector with norm 0 and in the opposite direction of u ( 5, ). Find a vector with same magnitude as u ( 5, ) but in the opposite direction of u 3. A pilot wants his groundspeed to be 60 mph on a bearing of 80. An east wind is blowing at 65 mph. What should his airspeed and heading be so that the wind will blow him back onto his intended course? 4. Find the volume of the parallelepiped whose edges are u ( 3,,6), v (4,,5), and w (,,7). 5. Points A(,5,6), B(5,,) and C(8,,6) are given. Find a vector perpendicular to the plane ABC going through point A. 6. Find the area of the parallelogram whose consecutive sides are the vectors ( 5,7,) and (3,4,) 7. Find the angle between the vectors ( 4,5,6) and (,,). 8. A plane takes off from Lake Arrowhead airport traveling 300 mph on a bearing of 00 degrees. A south wind is blowing (toward the north) at 5 miles per hour. Find the groundspeed and true course of the plane. Revised: 5/9/03
2 9. A pilot is flying at 68 mph. She wants her flight path to be on a bearing of 57. The wind is blowing from the north at 37. miles per hour. On what bearing should she fly to have the wind blow her plane on to course? 30. A plane is headed due south with an airspeed of 9 mph. A wind from a heading of 78 is blowing at 3 mph. Find the groundspeed and resulting bearing of the plane. Resolve the following forces into component form: lbs. at 0 above horizontal 3. 3 mph on a bearing of 0 Find: a) The resultant of F and F in component form. b) The magnitude and direction (as a bearing) of the resultant from (a) 33. F = (4, 5) F = (3,6) 34. F = ( 6, 30) F = (0, 4) Use the dot product formula u v u v cos to find the measure of the angle between u and v. 35. u = (4,, ) v = ( 5, 8, 6) 36. Find the angle between F and F in problem # u = (8, 3) v = (5, 7) Use the vector cross product to find the area of the triangle ABC whose vertices are given. (Area of parallelogram = u v ) Hint: Resolve AB and AC into component form and find the area of the parallelogram. 38. A( 8, 3, 0) B(,, 5) C(4, 9, ) 39. A(4,, 6) B (5,, 4) C(, 0, 6) Find the volume of the parallelepiped whose edges are the vectors u, v, and w 40. u = (3,, 3) v = (, 4, ) w = (,, ) Write the vector t as a linear combination of u (,5) and v (3, ) 3 4. t = (8, 7) 4. t =, 43. Find the vector with length 8 in the same direction as v = 7,,. Verify that its length is At what bearing and speed would a pilot head if he wants to fly due north at 345 mph when a 40 mph west wind is blowing? 45. A Major League baseball diamond is a square having 90 ft. sides. If the pitcher stands 60 feet 6 inches from home plate, how far is he from nd base? 46. Jim can swim at a rate of 3 mph. If he heads for a point directly across a river in which the current is 0 mph, by how many degrees does the direction in which he actually swims differ from his intended direction? If the river is 3 yards wide, will he make it across before reaching the falls that are yards downstream? 47. In a naval maneuver, two ships rendezvous at point A. One then proceeds east 0 miles and north 4 miles to point B. At what bearing should the second ship head to meet the first ship at point B? Use the formula Work = F d cos to find the work done (in joules) by the force F, with given magnitude and direction, in moving an object the given distance at the given angle. (F is the force vector, d is the direction vector, is the angle between F and d) 48. force of 5 N at 45 along a ramp 60 meters long at force of 9 N at 0 along a ramp 00 meters long at force of 5 N at 4 up a vertical cable a distance of 5 meter. 5. force of 4.36 N at 8 up a vertical cable a distance of meter. Draw a figure to solve each of the following problems on a inclined plane. 5. What is the weight of a car sitting on a 4 slope if the force required to push the car up the hill is 750 pounds.? 53. What is the force required to push a 40 pound lawn mower up a hill inclined at 8? Revised: 5/9/03
3 54. A 3000 pound car is sitting on a slope. Find the magnitude of the force required to push the car up the slope and the magnitude of the force holding the car on the slope? 55. Two men push a 00 pound box up a ramp. Each man pushes with a force of 75 pounds in the direction of the incline of the ramp, and the box just moves up the ramp. At what angle is the ramp inclined? 56. Find the amount of work done by a force F of N at 50 in moving a box of apples 30 meters up a ramp inclined at 0 Solve each of the following using vectors. 57. Two forces of 69 N and 43 N acts at a point. The resultant force is 786 N. Find the angle between the forces. 58. Three forces acting at a point are in equilibrium. The forces are 980 lbs, 760 lbs. and 0 lbs. Find the angles between the forces ( to the nearest minute) 59. A force of 76 lbs. Makes an angle of 7850 with a second force. The resultant of the two forces makes an angle of 40 with the first force. Find the magnitude of the second force and the resultant. 60. A force of 8.7 lbs. makes an angle of 40 with a second force. The resultant of the two forces makes an angle of 340 with the first force. Find the magnitude of the resultant. 6. A crate is supported by two ropes. One rope makes an angle of 460 with horizontal and has a tension of 89.6 lbs. on it. The other rope is horizontal. Find the weight of the crate and the tension in the horizontal rope. 6. Two people are carrying a box, one on each side of the box. One person exerts a force of 50 lbs. at an angle of 6.4 with horizontal. The other person exerts a force 4 lbs. at an angle of 54.9with horizontal. Find the weight of the box. 63. Two tugboats are pulling a disabled speed boat into port with forces of 40 lbs. ad 480 lbs. The angle between these forces is 8.. Find the direction and magnitude of the equilibrant. 64. If u ( 4,5) and v (4,0), find uv 65. If u (6,5), v ( 7, ) and t ( 3, 5), find 3u3( t v) 66. If P(,3, 4) and Q (7,0, ), find the midpoint of PQ 67. If P(,3, 4) and Q (7,0, ), find PQ 68. If A(0,0,), B(3,4,0) and C ( 4,3,0), is ABC isosceles, scalene or equilateral? Why? 69. If u (3, 4) and w(,), then find the norm of 3w u 70. Are the following vectors orthogonal? (3,4,) and (4,, 4) 7. Find the measure of the angle between ( 3, 3, 6) and (4,, ) to the nearest tenth. 7. Find k so that the given vectors are perpendicular: (5,8) and (, k) 73. Write the vector (6, 3) as a linear combinaon of i and j. 74. Write t ( 8, 9) as a linear combination of u (, ) and v (6,3) 75. Write t ( 3,) as a linear combination of u (,5) and v (5,) 76. Write t (, 9) as a linear combination of u (,3) and v (, 3) 77. Find a vector with norm 0 and in the opposite direction of u (,5) 78. Find the vector with same magnitude as u (,5) but in the opposite direction ofu For each of the following let u ( 8,6), v (5,), s (3, 4), w(,3,5), t (,, 4) 79. Find w 80. Find u 8. Find 3u v 8. Find u 83. Find v 7u 84. Find 85. wt 86. u v u 87. Are u and s orthogonal? Why or why not? 88. Find the measure of the angle between s and v to the nearest tenth. 89. Find w t Revised: 5/9/03
4 90. Find the area of the parallelogram with sides w and t 9. An airplane takes off on a bearing of 80 and airspeed of 350 mph. A north wind is blowing at 35 mph. Find the groundspeed and true course of the plane. 9. Two forces act on the same point. One is force of 30 N at 45. The other is a force of 40 N at 0. Find the magnitude and direction of the resultant force. 93. Three forces of 4 lbs, 0 lbs and 0 lbs are in equilibrium. Find the measure of the angle between the: A) 4 lb force and the 0 lb. force. B) 4 lb force and the 0 lb force C) 0 lb force and the 0 lb. force. 94. A pilot wants his groundspeed to be 60 mph on a bearing of 80. An east wind is blowing at 65 mph. What should his airspeed and heading be so that the wind will blow him back onto his intended course? 95. Find the volume of the parallelepiped whose edges are u ( 3,,6), v (4,,5), and w(,,7). 96. Points A(,5,6), B(5,,) and C(8,,6) are given. Find a vector perpendicular to the plane ABC going through point A. 97. Find the area of the parallelogram whose consecutive sides are the vectors ( 5,7,) and (3,4,) 98. Find the angle between the vectors ( 4,5,6) and (,,). Revised: 5/9/03
5 ANSWERS:. 35., ,4, , 3 3 4, No. dp = ,,,. j ,, ft i 3 j 8. t = 3 v t u v 0. t u v., mph 5. ( 5, 30, 33)., 5 (or any scalar multiple of this) mph, 95.4º º mph, 86.5º 3. (87.94, 68.40) 3. (305.8, 64.87) on a bearing of 73.44º on a bearing of º º º 73.9º units units units 3 4. t 3u4v 4. t v ,, mph, 353.4º ft. from nd base yards º joules joules joules joules lbs lbs. 54. Force to push the car up the hill: lbs. Force to keep the car on the º joules º hill: lbs º, 4.49º, 6.6º 59. F :89.59 lbs, r: 8.57 lbs lbs 6. W: 64.8 lbs T: 6.9 lbs lbs lbs 67.º from 480 lb force 64.6º from 40 lb force 64. (,0) 65. (30, 6) ,, isosceles orthogonal i 3j 74. t = 3v 75. t = 4 u 7 v t u v 77., (, 5) ( 6,) 8. ( 34, 6) , ( 7, 6,7) 93. a) 53. b) c) 54.5 Revised: 5/9/03
6 96. ( 5, 30, 33) (or any scalar multiple of this vector) Revised: 5/9/03
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