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Math252Sols1F08V5

# Math252Sols1F08V5 - QUIZ 1(CHAPTER 14 SOLUTIONS MATH 252...

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QUIZ 1 (CHAPTER 14) - SOLUTIONS MATH 252 – FALL 2008 – KUNIYUKI SCORED OUT OF 125 POINTS ± MULTIPLIED BY 0.84 ± 105% POSSIBLE Clearly mark vectors, as we have done in class. I will use boldface, but you don’t! When describing vectors, you may use either or “ i ± j ± k ” notation. Assume we are in our usual 2- and 3-dimensional Cartesian coordinate systems. Give exact answers, unless otherwise specified. 1) Consider the points P 1, 3 () and Q 5, 8 . Find the vector in V 2 that has the same direction as the vector [corresponding to] PQ ±² ±± and has length 100. (8 points) Let a be the vector corresponding to PQ : a = 5 ± 1, 8 ± 3 = 4, 5 Find the magnitude (or length) of a : a = 4, 5 = 4 2 + 5 2 = 41 Find the unit vector u in V 2 that has the same direction as a : u = a a = 4, 5 41 = 4 41 , 5 41 or 44 1 41 , 54 1 41 Multiply u by 100 to get the desired vector of magnitude 100: 100 u = 100 1 41 , 1 41 = 400 41 41 , 500 41 41

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2) Write an equation (in x , y , and z ) of the sphere with center 0, 4, ± 2 () that is tangent to the xy -plane in our usual three-dimensional xyz -coordinate system. (4 points) The center of the sphere has z -coordinate ± 2 , so it is two units below the xy -plane, and the radius of the sphere is 2. We use the template: x ± x 0 2 + y ± y 0 2 + z ± z 0 2 = r 2 . x ± 0 2 + y ± 4 2 + z ±± 2 2 = 2 2 x 2 + y ± 4 2 + z + 2 2 = 4 3) Assume that a , b , and c are nonzero vectors in V 3 . Prove: comp c a + b = comp c a + comp c b . You may use the comp formula and the basic dot product properties listed in the book without proof. (Writing a = a 1 , a 2 , a 3 and so forth will not help here.) (6 points) comp c a + b = a + b c c = a c + b c c = a c c + b c c = comp c a + comp c b 4) Write the Cauchy-Schwarz Inequality. Let a and b be vectors in V n , where n is some natural number. (4 points) a b ± a b
5) You do not have to show work for this problem. There are many possible answers for part a) and for part b). (6 points total; 3 points each) a) Assuming v = 2, 3 in V 2 , find a nonzero vector w in V 2 that is orthogonal to v . Find a nonzero vector w in V 2 such that v w = 0. Sample answer: 3, ± 2 . b) Assuming a = 3 i + 4 j + 5 k in V 3 , find a nonzero vector b in V 3 such that a ± b = 0 . Find a nonzero vector b in V 3 that is parallel to a . We can choose any nonzero scalar multiple of a , even a itself. Sample answer: a , or 3, 4, 5 . 6) Let a , b , and c be vectors in V 3 . (4 points total; 2 points each) a) a ± b () c is … (Box in one:) a scalar a vector neither, or undefined a ± b is a vector in V 3 , and the dot product of a vector in V 3 and a vector in V 3 is a scalar. We call a ± b c a triple scalar product (“TSP”).

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Math252Sols1F08V5 - QUIZ 1(CHAPTER 14 SOLUTIONS MATH 252...

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