Homework 1 Solution - MAT 267 Written Homework#1 10.1 10.2...

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MAT 267 Written Homework #1 SOLUTIONS 10.1, 10.2, 10.3, 10.4 Due: September 1 Solve the following problems, showing any necessary work. 1. [1 point] Find an equation for the sphere which contains the points (3 , - 1 , 4) and (1 , 3 , 2), and whose center is the midpoint of the line segment joining these points. Solution: The midpoint of the two given points is 3 + 1 2 , - 1 + 3 2 , 4 + 2 2 = (2 , 1 , 3) . The radius of the sphere is the distance between (2 , 1 , 3) (the center) and (3 , - 1 , 4) (a point on the sphere), which is r = p (2 - 3) 2 + (1 - ( - 1)) 2 + (3 - 4) 2 = 1 + 4 + 1 = 6 , so an equation for the sphere is ( x - 2) 2 + ( y - 1) 2 + ( z - 3) 2 = ( 6) 2 . 2. [2 points] Let ~u = h 2 , 4 i and ~v = - ~ ı + 2 ~ . Find ~u + ~v , 3 2 ~u , | ~u | , ~u · ~v , and proj ~v ( ~u ). Solution: ~u + ~v = h 2 + ( - 1) , 4 + 2 i = h 1 , 6 i 3 2 ~u = 3 2 · 2 , 3 2 · 4 = h 3 , 6 i | ~u | = p 2 2 + 4 2 = 20 ~u · ~v = 2 · ( - 1) + 4 · 2 = 6 proj ~v ( ~u ) = ~u · ~v ~v · ~v ~v = 6 5 · h- 1 , 2 i = - 6 5 , 12 5 3. [2 points] Let ~u = 2 ~
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Unformatted text preview: ~v , ~u × ~v , the angle between ~u and ~v , and the vector of length 1 that points in the same direction as ~v . Solution: ~u · ~v = 2 · 0 + (-1) · 2 + 3 · 3 = 7 ~u × ~v = µ µ µ µ µ µ ~ ı ~ ~κ 2-1 3 2 3 µ µ µ µ µ µ = h-9 ,-6 , 4 i To find the angle between ~u and ~v , use the formula ~u · ~v = | ~u | · | ~v | · cos θ : θ = cos-1 ± ~u · ~v √ ~u · ~u · √ ~v · ~v ² = cos-1 ± 7 √ 14 · √ 13 ² = cos-1 (0 . 5188745215) , which is 1 . 025262482 radians or 58 . 74321311 ◦ . The vector of length 1 that points in the same direction as ~v is ~v | ~v | = h , 2 , 3 i √ 2 + 2 2 + 3 2 = ³ , 2 √ 13 , 3 √ 13 ´ ....
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