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prac1bsol

# prac1bsol - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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v v v v v v v v r a A p P 18.02 Practice Exam 1 B Solutions Problem 1. a) P = (1 , 0 , 0), Q = (0 , 2 , 0) and R = QP = ˆ ı QR = + 3 k ˆ . (0 , 0 , 3). Therefore −−→ and −−→ QP QR b) cos θ = v −−→ v · v −−→ v = a 1 , 2 , 0 A · a 0 , 2 , 3 A = 4 v QP vv QP v 1 2 + 2 2 2 2 + 3 2 65 v −−→ vv −−→ v Problem 2. a) −−→ = a− 1 , 2 , 0 A , −→ = a− 1 , 0 , 3 A . v v PQ PR v ˆ ı ˆ k ˆ v −−→ PQ × −→ = v 1 0 v = 6ˆ + 2 ˆ PR 2 ı + 3ˆ k . 1 0 3 Then area (Δ) = 2 1 v v PQ −−→ × PR −→ v v = 2 1 6 2 + 3 2 + 2 2 = 2 1 49 = 2 7 . b) A normal to the plane is given by −→ = PQ × PR a 6 , 3 , 2 A . N −−→ −→ = Hence the equation has the form 6 x + 3 y + 2 z = d . Since P
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prac1bsol - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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