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Calc 3 Exam 1 Sin 2011 - MAC2313 Exam 1 Dr Sin No...

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MAC2313 Exam 1 Dr Sin No Calculators. Answer the questions in the spaces provided on the question sheets. Please write your answers in full detail. If you run out of room for an answer, continue on the back of the page or on other paper. Name: 1. (7 points) Sketch the surface give by x 2 + y 2 - z 2 = 4 . (Start by sketching some traces (intersections with planes) and use them to help you see the 3-D picture.) Solution: Figure 1: z = 0 trace
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MAC2313 Exam 1 Dr Sin Figure 2: x = 0 trace ( y = 0 trace looks the same) Figure 3: 3D picture
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MAC2313 Exam 1 Dr Sin 2. (8 points) (a) (4 points) Find the angle between the planes 5 x - y - 2 z = 1 and 4 x + y + z = 6. (You may leave your answer in terms of inverse trig functions.) Solution: We can take n 1 = h 5 , - 1 , - 2 i and n 2 = h 4 , 1 , 1 i as normal vectors of the planes. Then if θ is the angle between the planes, we have cos θ = n 1 · n 2 | n 1 || n 2 | = 17 30 18 . so θ = cos - 1 ( 17 30 18 ). (b) (4 points) Find the distance from the point (1 , 0 , 1) to the plane x + y + z = 5.
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