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prac1Asol

# prac1Asol - 18.02 Practice Exam 1 A Solutions Problem 1 1 i...

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18.02 Practice Exam 1 A – Solutions Problem 1. a) --→ OQ = ˆ ı + ˆ  + ˆ k; --→ OR = 1 2 ˆ ı + ˆ  + 1 2 ˆ k. b) cos θ = --→ OQ · --→ OR | --→ OQ || --→ OR | = h 1 , 1 , 1 i·h 1 2 , 1 , 1 2 i 3 q 3 2 = 2 2 3 . Problem 2. Velocity: ~ V = d ~ R dt = h- 3 sin t, 3 cos t, 1 i . Speed: | ~ V | = q 9 sin 2 t + 9 cos 2 t + 1 = 10. Problem 3. a) Minors: 1 1 2 - 2 - 2 - 2 - 3 - 5 - 6 . Cofactors: 1 - 1 2 2 - 2 2 - 3 5 - 6 . Inverse: 1 2 1 2 - 3 - 1 - 2 5 2 2 - 6 . b) X = A - 1 B = - 3 4 - 4 . Problem 4. Q = top of the ladder: --→ OQ = h 0 , L sin θ i ; R = bottom of the ladder: --→ OR = h- L cos θ, 0 i . Midpoint: --→ OP = 1 2 ( --→ OQ + --→ OR ) = h- L 2 cos θ, L 2 sin θ i . Parametric equations: x = - L 2 cos θ , y = L 2 sin θ . Problem 5. a) ---→ P 0 P 1 × ---→ P 0 P 2 = f f f f f f f ˆ ı ˆ ˆ k - 1 - 1 1 0 - 2 1 f f f f f f f = ˆ ı + ˆ  + 2 ˆ k. Area = 1 2 | ---→ P 0 P 1 × ---→ P 0 P 2 | = 1 2 6. b) Normal vector:
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