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Unformatted text preview: Review with Solution for Midterm I, MATH 20C, 2009 1 Chapter 12. 1. [12.2] Find an equation of a sphere if one of its diameters has endpoints (2 , 1 , 4) and (4 , 3 , 10). Ans. The center of the sphere is the midpoint of the given two end points of one diameter, i.e., center = 2 + 4 2 , 1 + 3 2 , 4 + 10 2 = (3 , 2 , 7) . The radius of the sphere is the distance between either one of the end points of the given diameter and the center of the sphere ; radius = p (3 2) 2 + (2 1) 2 + (7 4) 2 = √ 11 . Therefore, the sphere equation is ( x 3) 2 + ( y 2) 2 + ( z 7) 2 = 11 . 2. [12.2, 12.3, 12.4] Given P = (2 , 1 , 3), Q = (1 , 1 , 0), R = (1 , 1 , 1), S = (0 , 1 , 2) determine whether the vectors→ PQ and→ RS (a) are parallel (b) are perpendicular by using the cross and dot products. Ans. (a)→ PQ = < 1 , 2 , 3 > ,→ RS = < 1 , , 1 > . Note that two vectors are parallel if and only if the cross product of them is zero.→ PQ ×→ RS = i j k 1 2 3 1 1 = 2 i ( 2) j 2 k = 2 i + 2 j 2 k 6 = 0 . Hence, they are not parallel. (b) Also, note that two vectors are perpendicular if and only if the dot product of two vectors is zero.→ PQ •→ RS = ( 1) · ( 1) + ( 2) · 0 + ( 3) · ( 1) = 1 + 3 = 4 6 = 0 . Hence, they are not perpendicular. 3. [12.3, 12.4] Let a = < 2 , 2 , 2 > and b = < 5 ,c, 5 > be vectors, where c is to be determined. (a) For what value of c is b = < 5 ,c, 5 > orthogonal to a = < 2 , 2 , 2 > ? (b) For what value of c is b = < 5 ,c, 5 > parallel to a = < 2 , 2 , 2 > ? Ans. (a) a • b = 10 + 2 c + 10 = 0, if c = 10. So they are perpendicular when c = 10. (b) Parallel means a = λ b for some constant λ . This means that we must have 2 = 5 λ , 2 = cλ . From the first equation, we get λ = 2 / 5 and plugging this in to the second gives c = 5. 4. [12.4] Let A = (2 , 1 , 1), B = (3 , , 2), C = (3 , 2 , 1) and D = ( 2 , , 1). (a) Find the area of the parallelogram that has AB and AC as adjacent sides. (b) Find the volume of the parallelepiped that has edges AB,AC , and AD . Ans. (a) Let u = < 1 , 1 , 1 > be the vector from A to B and let v = < 1 , 1 , 2 > be the vector from A to C . Then Area =  u × v  =  < 1 , 3 , 2 >  = √ 1 + 9 + 4 = √ 14 . (b) Let w = < 4 , 1 , 2 > be the vector from A to D . Then Volume =  ( u × v ) • w  =  < 1 , 3 , 2 > • < 4 , 1 , 2 >  =  4 + 3 + 4  = 11 where u and v are the same as in the solution to part (a). 5. Let a = < 1 , 1 , 1 > and b = < 2 , 2 , 1 > . (a) [12.4] Find two vectors that are orthogonal to both a and b ....
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 Fall '08
 Lunasin
 Math

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