Math252Sols1F07

Math252Sols1F07 - QUIZ 1(CHAPTER 14 SOLUTIONS MATH 252 FALL...

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QUIZ 1 (CHAPTER 14) - SOLUTIONS MATH 252 – FALL 2007 – 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) Assume that a 1 , a 2 , p , and q are real numbers. Prove that, if a = a 1 , a 2 , then p + q ( ) a = p a + q a . Show all steps! (10 points) Let a = a 1 , a 2 . p + q ( ) a = p + q ( ) a 1 , a 2 = p + q ( ) a 1 , p + q ( ) a 2 = pa 1 + qa 1 , pa 2 + qa 2 = pa 1 , pa 2 + qa 1 , qa 2 = p a 1 , a 2 + q a 1 , a 2 = p a + q a Q.E.D. Note: Many people who scored 4 points did the following: p + q ( ) a = p + q ( ) a 1 , a 2 = p a 1 , a 2 + q a 1 , a 2 You're using the property you're trying to prove! That's circular reasoning! = p a + q a 2) Write an inequality in x , y , and/or z whose graph in our usual three-dimensional xyz -coordinate system consists of the sphere of radius 4 centered at the origin and all points inside that sphere. (4 points) The equation of the sphere of radius 4 centered at the origin: x 2 + y 2 + z 2 = 16 . We also want all points inside that sphere, so our inequality is: x 2 + y 2 + z 2 16 . Another approach: We want all points whose distance from the origin is at most 4.

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3) Find all real values of c such that the vectors c i + 10 j + c k and c i 2 j k are orthogonal. (8 points) The vectors are orthogonal Their dot product is 0. c i + 10 j + c k ( ) c i 2 j k ( ) = c ,10, c c , 2, 1 = c 2 20 c = c 2 c 20 Find the real zeros: c 2 c 20 = 0 c + 4 ( ) c 5 ( ) = 0 c = 4 or c = 5 4) Assume that a and b are vectors in V n , where n is some natural number. Using entirely mathematical notation (i.e., don’t use words) … (8 points; 4 points each) a) Write the Cauchy-Schwarz Inequality. a b a b b) Write the Triangle Inequality. a + b a + b 5) 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 scalar, and a scalar times a vector is a vector. Think: Scalar multiplication. b) a × b × c ( ) is … (Box in one:) a scalar a vector neither, or undefined In fact, this is a Triple Vector Product. b × c is a vector, and a crossed with it is a vector.
6) Assume that a , b , and c are three nonzero vectors in V 3 such that a × b = a × c . Which of the following must be true? Box in one: (3 points)

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This note was uploaded on 09/08/2011 for the course MATH 252 taught by Professor Staff during the Spring '11 term at Mesa CC.

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Math252Sols1F07 - QUIZ 1(CHAPTER 14 SOLUTIONS MATH 252 FALL...

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