# HW7S - Gauthier, Joseph Homework 7 Due: Oct 11 2007, 3:00...

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Gauthier, Joseph – Homework 7 – Due: Oct 11 2007, 3:00 am – Inst: JEGilbert 1 This print-out should have 17 questions. Multiple-choice questions may continue on the next column or page – fnd all choices beFore answering. The due time is Central time. 001 (part 1 oF 1) 10 points Which oF the Following statements are true For all lines and planes in 3-space? I. two lines perpendicular to a third line are parallel , II. two lines perpendicular to a plane are parallel , III. two planes parallel to a third plane are parallel . 1. II and III only correct 2. all oF them 3. none oF them 4. III only 5. I and III only 6. I and II only 7. II only 8. I only Explanation: I. ±ALSE: the x -axis and y -axis are perpen- dicular, not parallel, yet both are perpendic- ular to the z -axis. 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. keywords: 3-space, lines, planes, parallel, T/±, perpendicular, 002 (part 1 oF 1) 10 points Which oF the Following surFaces is the graph oF 4 x + 3 y + 6 z = 12 in the frst octant? 1. x y z cor- rect 2. x y z 3. x y z

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Gauthier, Joseph – Homework 7 – Due: Oct 11 2007, 3:00 am – Inst: JEGilbert 2 4. x y z 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 4 x + 3 y + 6 z = 12 . Now the x -intercept occurs at y = z = 0, i.e. at x = 3; similarly, the y -intercept is at y = 4, while the z -intercept is at z = 2. By inspection, therefore, the graph is x y z keywords: 3Dgraph, plane, linear equation, intercepts, 003 (part 1 of 1) 10 points Which equation has the surface x y z x y z as its graph? 1. x + y = 1 2. y - x = 1 correct 3. x + z = 1 4. y + z = 1 5. z - y = 1 6. z - x = 1 Explanation: As the surface is a plane, the equation must be linear; in addition, for Fxed y and varying x , the corresponding value of z stays constant. So the equation must depend only on y and z . To determine the equation precisely, we
Gauthier, Joseph – Homework 7 – Due: Oct 11 2007, 3:00 am – Inst: JEGilbert 3 thus look at the trace of the surface on the yz -plane as shown in Consequently, the linear equation is y - x = 1 . keywords: 3Dgraph, linear equation, trace, plane 004 (part 1 of 1) 10 points Find parametric equations for the line pass- ing through the point P (3 , - 1 , 3) and parallel to the vector h 2 , 4 , - 2 i . 1. x = 3 - 2 t, y = 1 - 4 t, z = 3 - 2 t 2. x = 2 + 3 t, y = 4 - t, z = - 2 + 3 t 3. x = 3 + 2 t, y = - 1 + 4 t, z = 3 - 2 t correct 4. x = - 3 + 2 t, y = 1 + 4 t, z = - 3 - 2 t 5. x = 2 + 3 t, y = 4 + t, z = 2 - 3 t 6. x = 2 - 3 t, y = - 4 + t, z = - 2 + 3 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

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## HW7S - Gauthier, Joseph Homework 7 Due: Oct 11 2007, 3:00...

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