M408D Quest Homework 8-solutions

M408D Quest Homework 8-solutions - hernandez (ah29758)...

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Unformatted text preview: hernandez (ah29758) M408D Quest Homework 8 pascaleff (54550) 1 This print-out should have 12 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points Find parametric equations for the line passing through the points P (4 , 1 , 2) and Q (2 , 5 , 3). 1. x = 2 4 t, y = 4 + t, z = 1 + 2 t 2. x = 4 + 2 t, y = 1 + 4 t, z = 2 t 3. x = 2 + 4 t, y = 4 + t, z = 1 2 t 4. x = 4 2 t, y = 1 + 4 t, z = 2 + t correct 5. x = 4 2 t, y = 1 4 t, z = 2 + t 6. x = 2 4 t, y = 4 t, z = 1 + 2 t Explanation: A line passing through a point P ( a, b, c ) and having direction vector v is given para- metrically by r ( t ) = a + t v , a = ( a, b, c ) . Now PQ = ( 2 , 4 , 1 ) is a direction vector for the given line, so a = ( 4 , 1 , 2 ) , v = ( 2 , 4 , 1 ) . Thus r ( t ) = ( 4 2 t, 1 + 4 t, 2 + t ) . Consequently, x = 4 2 t, y = 1 + 4 t, z = 2 + t are parametric equations for the line. 002 10.0 points Find an equation for the plane passing through the origin and parallel to the plane x + 6 y 5 z = 2 . 1. 5 x y + 6 z = 0 2. 6 x + 5 y + 6 z = 0 3. 5 x + y 6 z + 2 = 0 4. 6 x 5 y 6 z + 2 = 0 5. x + 6 y 5 z = 0 correct 6. x 6 y + 5 z + 2 = 0 Explanation: The scalar equation for the plane through P ( a, b, c ) with normal vector n = A i + B j + C k is A ( x a ) + B ( y b ) + C ( z c ) = 0 . But P ( a, b, c ) = (0 , , 0) if the plane passes through the origin, while n = i + 6 j 5 k if the plane is parallel to x + 6 y 5 z = 2 since parallel planes have the same normal vector. Consequently, the plane has equation x + 6 y 5 z = 0 . 003 10.0 points Find parametric equations for the line pass- ing through the point P (2 , 1 , 4) and perpen- dicular to the plane 3 x + 4 y z = 5 . 1. x = 3 + 2 t, y = 4 + t, z = 1 4 t 2. x = 3 + 2 t, y = 4 t, z = 1 + 4 t hernandez (ah29758) M408D Quest Homework 8 pascaleff (54550) 2 3. x = 2 + 3 t, y = 1 + 4 t, z = 4 t correct 4. x = 2 + 3 t, y = 1 + 4 t, z = 4 t 5. x = 2 3 t, y = 1 4 t, z = 4 t 6. x = 3 2 t, y = 4 + t, z = 1 + 4 t Explanation: A line passing through a point P ( a, b, c ) and having direction vector v is given para- metrically by...
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M408D Quest Homework 8-solutions - hernandez (ah29758)...

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