HW 08-solutions - suleimenov (bs26835) HW 08 rusin (55565)...

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Unformatted text preview: suleimenov (bs26835) HW 08 rusin (55565) 1 This print-out should have 27 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. Note: here, the trace of a graph in a plane just means the intersection of the graph with the plane. (I dont know why they use that language.) 001 10.0 points The distance d from P to the line in Q R D d a b P is given by d = | a b | | a | where Q, R are points on and a = QR , b = QP . Use this formula to find the distance from P (1 , 4 , 7) to the line specified in parametric form by x = 3 + 2 t , y = 2 t , z = 5 + 2 t . 1. distance = 103 2 2. distance = 104 3 correct 3. distance = 103 3 4. distance = 105 3 5. distance = 104 2 6. distance = 105 2 Explanation: It is convenient to take Q = ( x (0) , y (0) , z (0)) = (3 , 2 , 5) and R = ( x (1) , y (1) , z (1)) = (5 , 1 , 7) . With this choice of Q and R , a = ( 2 , 1 , 2 ) , b = ( 2 , 2 , 2 ) , while a b = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle i j k 2 1 2 2 2 2 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = vextendsingle vextendsingle vextendsingle vextendsingle 1 2 2 2 vextendsingle vextendsingle vextendsingle vextendsingle i vextendsingle vextendsingle vextendsingle vextendsingle 2 2 2 2 vextendsingle vextendsingle vextendsingle vextendsingle j + vextendsingle vextendsingle vextendsingle vextendsingle 2 1 2 2 vextendsingle vextendsingle vextendsingle vextendsingle k = 6 i 8 j + 2 k . Consequently, d = | 6 i 8 j + 2 k | |( 2 , 1 , 2 )| = 104 3 . keywords: distance, distance from line, cross product, vector product, vector, length, 002 10.0 points For which of the following quadratic rela- tions is its graph a ellipsoid? 1. z 2 x 2 y 2 = 1 2. 2 x 2 + y 2 + 3 z 2 = 1 correct 3. z = x 2 + y 2 4. z = y 2 x 2 suleimenov (bs26835) HW 08 rusin (55565) 2 5. z 2 = x 2 + y 2 6. x 2 + y 2 z 2 = 1 Explanation: The graphs of each of the given quadratic relations is a quadric surface in standard posi- tion. We have to decide which quadric surface goes with which equation - a good way of do- ing that is by taking plane slices parallel to the coordinate planes, i.e. , by setting respec- tively x = a , y = a and z = a in the equations once weve decided what the graphs of those plane slices should be. Now as slicing of shows, a slice of this ellipsoid by each of x = a , y = a and z = a will be an ellipse (or circle) so long as the slice intersects the ellipsoid. Only the graph of 2 x 2 + y 2 + 3 z 2 = 1 has this property....
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HW 08-solutions - suleimenov (bs26835) HW 08 rusin (55565)...

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