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HW09-solutions

# HW09-solutions - zakaria(mmz255 HW09 gilbert(55485 This...

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zakaria (mmz255) – HW09 – gilbert – (55485) 1 This print-out should have 19 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0points Which of the following statements are true for all lines and planes in 3-space? I. two lines parallel to a third line are par- allel , II. two lines perpendicular to a plane are parallel , III. two planes parallelto a third plane are parallel . 1. I and III only 2. all of them correct 3. III only 4. II and III only 5. I only 6. none of them 7. II only 8. I and II only Explanation: I. TRUE: each of the two lines has a direc- tion vector parallel to the direction vector of the third line, so must be scalar multiples of each other. II. TRUE: the two lines will have direction vectors parallel to the normal vector of the plane, and so be parallel, hence the two lines are parallel. III. TRUE: each of the two planes has a normal vector parallel to the normal vector of the third plane, and so are parallel, hence the planes are parallel. 002 10.0points Which of the following surfaces is the graph of 6 x + 3 y + 4 z = 12 in the first octant? 1. x y z 2. x y z 3. x y z correct 4. x y z

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zakaria (mmz255) – HW09 – gilbert – (55485) 2 5. x y z 6. x y z Explanation: Since the equation is linear, it’s graph will be a plane. To determine which plane, we have only to compute the intercepts of 6 x + 3 y + 4 z = 12 . Now the x -intercept occurs at y = z = 0, i.e. at x = 2; similarly, the y -intercept is at y = 4, while the z -intercept is at z = 3. By inspection, therefore, the graph is x y z 003 10.0points Find parametric equations for the line pass- ing through the point P (4 , 4 , 1) and parallel to the vector ( 1 , 1 , 4 ) . 1. x = 4 + t, y = 4 + t, z = 1 4 t correct 2. x = 4 t, y = 4 t, z = 1 4 t 3. x = 1 + 4 t, y = 1 + 4 t, z = 4 t 4. x = 1 4 t, y = 1 + 4 t, z = 4 + t 5. x = 4 + t, y = 4 + t, z = 1 4 t 6. x = 1 + 4 t, y = 1 4 t, z = 4 + 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 for the given line, a = ( 4 , 4 , 1 ) , v = ( 1 , 1 , 4 ) . Thus r ( t ) = ( 4 + t, 4 + t, 1 4 t ) . Consequently, x = 4 + t, y = 4 + t, z = 1 4 t are parametric equations for the line. keywords: line, parametric equations, direc- tion vector, point on line 004 10.0points A line passes through the point P (4 , 3 , 2) and is perpendicular to the plane x + y 2 z = 2 . At what point Q does intersect the xy - plane? 1. Q (5 , 4 , 0) correct 2. Q (4 , 5 , 0) 3. Q (3 , 2 , 0)
zakaria (mmz255) – HW09 – gilbert – (55485) 3 4. Q (3 , 0 , 4) 5. Q (2 , 0 , 3) 6. Q (5 , 0 , 2) Explanation: Since the xy -plane is given by z = 0, we have to find an equation for and then set z = 0. Now a line passing through a point P ( a, b, c ) and having direction vector v is given parametrically by r ( t ) = a + t v , a = ( a, b, c ) . But for , its direction vector will be parallel to the normal to the plane x + y 2 z = 2 .

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HW09-solutions - zakaria(mmz255 HW09 gilbert(55485 This...

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