HW08-solutions

# HW08-solutions - moore(jwm2685 HW08 gilbert(55485 This...

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moore (jwm2685) – HW08 – gilbert – (55485) 1 This print-out should have 23 questions. Multiple-choice questions may continue on the next column or page – fnd all choices beFore answering. 001 10.0 points Suppose you start at the origin in 3-space, move along the x -axis a distance oF 3 units in the positive direction, and then move down- ward a distance oF 2 units. What are the coordinates oF your position? 1. (0 , 3 , 2) 2. (3 , 0 , 2) correct 3. (3 , 0 , 2) 4. (3 , 2 , 0) 5. (0 , 2 , 3) Explanation: We start at the origin, which has coordi- nates (0 , 0 , 0). ±irst we move 3 units along the positive x -axis, a²ecting only the x - coordinate, bringing us to the point (3 , 0 , 0). We then move 2 units downward, in the nega- tive z -direction. Thus only the z -coordinate is a²ected, and so we arrive at the point having coordinates (3 , 0 , 2) . keywords: 3-space, coordinates, 002 10.0 points Which one oF the points P ( 1 , 9 , 8) , Q (3 , 1 , 0) , R (5 , 0 , 9) in 3-space is closest to the yz -plane? 1. P ( 1 , 9 , 8) correct 2. Q (3 , 1 , 0) 3. R (5 , 0 , 9) Explanation: The distance oF a point ( a, b, c ) in 3-space From the -plane is given by | a | . Conse- quently, oF the three points P ( 1 , 9 , 8) , Q (3 , 1 , 0) , R (5 , 0 , 9) the one closest to the -plane is P ( 1 , 9 , 8) . keywords: plane, distance in 3-space, 003 10.0 points A rectangular box is constructed in 3-space with one corner at the origin and other ver- tices at (4 , 0 , 0) , (0 , 3 , 0) , (0 , 0 , 5) . ±ind the length oF the diagonal oF the box. 1. length = 43 2. length = 34 3. length = 34 4. length = 5 2 correct 5. length = 50 6. length = 43 Explanation: We have to fnd the length oF BD in the fgure O D A B G F C E

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moore (jwm2685) – HW08 – gilbert – (55485) 2 given that OA = 4 , OC = 3 , OD = 5 . Now by Pythagoras’ theorem, length OB = length AC = 5 . But then, again by Pythagoras, length BD = 5 2 . Consequently, length = 5 2 . keywords: length diagonal, rectangular solid, Pythagoras’ theorem, ThreeDimSys, 004 10.0 points Determine the distance of the point Q (5 , 2 , 1) from the yz -coordinate plane. 1. distance = 1 2. distance = 2 3. distance = 30 4. distance = 5 correct 5. distance = 5 6. distance = 29 7. distance = 26 Explanation: Since the distance of a point P ( x, y, z ) from the xy -, -, and zx -coordinate planes is given respectively by | z | , | x | and | y | , the point Q (5 , 2 , 1) has distance = 5 from the -plane. keywords: coordinate plane, projection, point, distance, 3-space 005 10.0 points Determine the distance of the point Q (5 , 3 , 2) from the y -axis. 1. distance = 2 2. distance = 5 3. distance = 3 4. distance = 13 5. distance = 38 6. distance = 29 correct 7. distance = 34 Explanation: Since the distance of a point P ( x, y, z ) from the z -, x -, and y -axes is given respec- tively by r x 2 + y 2 , r y 2 + z 2 , r z 2 + x 2 , the point Q (5 , 3 , 2) has distance = 29 from the y -axis. keywords: coordinate plane, projection, point, distance, 3-space 006 10.0 points A triangle Δ PQR has vertices P ( 4 , 4 , 5) , Q (0 , 2 , 5) , R ( 2 , 1 , 0) .
moore (jwm2685) – HW08 – gilbert – (55485) 3 Use the distance formula to decide which one of the following properties the triangle has.

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