Math252SolsFinalF08

# Math252SolsFinalF08 - SOLUTIONS TO THE FINAL MATH 252 FALL...

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SOLUTIONS TO THE FINAL MATH 252 – FALL 2008 – KUNIYUKI 60 POINTS TOTAL (15 PROBLEMS; 4 POINTS EACH) No books allowed. An appropriate sheet of notes and a scientific calculator are allowed. 1) If a and b are vectors in V 2 such that a b < 0 , which of the following is true? Box in the best answer: a) The angle between a and b is in 0 ± ,90 ± ± ² ) . b) The angle between a and b is in 90 ± ,180 ± ( ± ² . c) a b tells us nothing about the angle between a and b . cos ± = a b a b and a b < 0 ² cos < 0 ² is in 90 ± , 180 ± ( ³ ´ . 2) Write an equation for the plane in xyz -space that is parallel to the plane with equation 3 x ± 2 y + 7 z = 10 and that contains the point 4, ± 1, 8 () . The given plane and the desired plane both have 3, ± 2, 7 as a normal vector. 3 x ± 4 ± 2 y ±± 1 + 7 z ± 8 = 0o r 3 x ± 2 y + 7 z ± 70 = r 3 x ± 2 y + 7 z = 70 3) Consider the graph of 4 x 2 ± 5 y 2 ± 6 z 2 = 3 in xyz -space. The trace of the graph in the plane x = 10 is … (Box in one): a) An Ellipse b) A Hyperbola c) A Parabola (See next page.)

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The trace is given by: 4 x 2 ± 5 y 2 ± 6 z 2 = 3, x = 10 ² 410 () 2 ± 5 y 2 ± 6 z 2 = x = 10 397 = 5 y 2 + 6 z 2 , x = 10 This is an ellipse. The ellipse family of traces makes sense, since planes of the form x = k are perpendicular to the axis of the similar graph of x 2 ± y 2 ± z 2 = 1, which, like the given
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Math252SolsFinalF08 - SOLUTIONS TO THE FINAL MATH 252 FALL...

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