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 · N | N | = h- 19 , 1 , 1 i · h 1 , 2 , 3 i √ 14 = 14 √ 14 = √ 14 . (You should draw the picture.) b) Line: P + t N = (20 + t, 2 t, 3 t ) ⇔ 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 ⇒ 14 t =- 14 ⇒ t =- 1 . ⇒ R = (19 ,- 2 ,- 3) . d) Let θ = ∠ PQR : | QP || QR | cos θ = QP · QR = h 19 ,- 1 ,- 1 i · h 18 ,- 3 ,- 4 i = 349 ⇒ cos θ = 349 √ 363 √ 349 ≈ . 98 ⇒ θ ≈ . 2 radians ≈ 11 ◦ . e) RP = h 1 , 2 , 3 i ⇒ | RP | = √ 14 . (Same as part (a).) f) Area = 1 2 | PQ × PR | . PQ × PR = i j k- 19 1 1- 1- 2- 3 =- i- 58 j + 39 k ⇒ area = 1 2 √ 1 + 58 2 + 39 2 = 1 2 √ 4886 = 34 . 9 . Problem 2. Parametrize by θ : r ( θ ) = OP = OQ + QP . OQ = h 2cos θ, 2sin θ i , QP is vertical of length 2 θ ⇒ QP = h , 2 θ i . ⇒ r ( θ ) = h 2cos θ, 2sin θ + 2 θ i . Physically this makes sense for 0 ≤ θ ≤ π. O θ 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 ⇒ a × v = h- 15 , , 20 i ⇒ κ = | a × v | | v | 3 = 1 5 . b) r · h 3 , ,- 4 i = 12sin t- 12sin t = 0 ⇒ they’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 ⇒ y = 1- 2 x 2 . Coordinates: A =(1 ,- 1) , B =( √ 2 / 2 , 0) , C =(0 , 1) , D =(- √ 2 / 2 , 0) , E =(- 1 ,- 1) . Path: Pos.: C → B → A → B → C → D → E → D → C (repeat) time: π 4 π 2 3 π 4 π 5 π 4 3 π 2 7 π 4 2 π x y E • D • C • B • A • (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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