Practice_Mid-Term_Solutions-Fall09

# Practice_Mid-Term_Solutions-Fall09 - 18.02a Practice...

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Unformatted text preview: 18.02a Practice Midterm Questions, Fall 2009 Solutions Problem 1. a) Distance = PQ Â· N | N | = h- 19 , 1 , 1 i Â· h 1 , 2 , 3 i âˆš 14 = 14 âˆš 14 = âˆš 14 . (You should draw the picture.) b) Line: P + t N = (20 + t, 2 t, 3 t ) â‡” x = 20 + t ; y = 2 t ; z = 3 t. c) Put equation for line into equation of plane: (20 + t ) + 4 t + 9 t = 6 â‡’ 14 t =- 14 â‡’ t =- 1 . â‡’ R = (19 ,- 2 ,- 3) . d) Let Î¸ = âˆ  PQR : | QP || QR | cos Î¸ = QP Â· QR = h 19 ,- 1 ,- 1 i Â· h 18 ,- 3 ,- 4 i = 349 â‡’ cos Î¸ = 349 âˆš 363 âˆš 349 â‰ˆ . 98 â‡’ Î¸ â‰ˆ . 2 radians â‰ˆ 11 â—¦ . e) RP = h 1 , 2 , 3 i â‡’ | RP | = âˆš 14 . (Same as part (a).) f) Area = 1 2 | PQ Ã— PR | . PQ Ã— PR = i j k- 19 1 1- 1- 2- 3 =- i- 58 j + 39 k â‡’ area = 1 2 âˆš 1 + 58 2 + 39 2 = 1 2 âˆš 4886 = 34 . 9 . Problem 2. Parametrize by Î¸ : r ( Î¸ ) = OP = OQ + QP . OQ = h 2cos Î¸, 2sin Î¸ i , QP is vertical of length 2 Î¸ â‡’ QP = h , 2 Î¸ i . â‡’ r ( Î¸ ) = h 2cos Î¸, 2sin Î¸ + 2 Î¸ i . Physically this makes sense for 0 â‰¤ Î¸ â‰¤ Ï€. O Î¸ Q P Problem 3. a) v ( t ) = h 4cos t,- 5sin t, 3cos t i . ds dt = | v | = 5 . T = v | v | = h 4 5 cos t,- sin t, 3 5 cos t i . a = h- 4sin t,- 5cos t,- 3sin t i â‡’ a Ã— v = h- 15 , , 20 i â‡’ Îº = | a Ã— v | | v | 3 = 1 5 . b) r Â· h 3 , ,- 4 i = 12sin t- 12sin t = 0 â‡’ theyâ€™re perpendicular. This means P moves in a plane with normal h 3 , ,- 4 i . Problem 4. x = sin t, y = cos2 t = cos 2 t- sin 2 t = 1- 2sin 2 t â‡’ y = 1- 2 x 2 . Coordinates: A =(1 ,- 1) , B =( âˆš 2 / 2 , 0) , C =(0 , 1) , D =(- âˆš 2 / 2 , 0) , E =(- 1 ,- 1) . Path: Pos.: C â†’ B â†’ A â†’ B â†’ C â†’ D â†’ E â†’ D â†’ C (repeat) time: Ï€ 4 Ï€ 2 3 Ï€ 4 Ï€ 5 Ï€ 4 3 Ï€ 2 7 Ï€ 4 2 Ï€ x y E â€¢ D â€¢ C â€¢ B â€¢ A â€¢ (continued) 1 18.02a Practice Midterm Questions, Fall 2009 Solutions 2 Problem 5....
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Practice_Mid-Term_Solutions-Fall09 - 18.02a Practice...

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