HW_7_Key - Miller, Kierste Homework 7 Due: Oct 11 2007,...

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Miller, Kierste – 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 plane are parallel , II. two planes parallel to a third plane are parallel , III. two lines perpendicular to a third line are parallel . 1. I and II only correct 2. I and III only 3. I only 4. none oF them 5. II and III only 6. II only 7. III only 8. all oF them Explanation: I. 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. II. 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. III. ±ALSE: the x -axis and y -axis are per- pendicular, not parallel, yet both are perpen- dicular to the z -axis. 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 + 6 y + 3 z = 12 in the frst octant? 1. x y z 2. x y z 3. x y z 4. x y z
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Miller, Kierste – Homework 7 – Due: Oct 11 2007, 3:00 am – Inst: JEGilbert 2 5. x y z 6. x y z correct 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 + 6 y + 3 z = 12 . Now the x -intercept occurs at y = z = 0, i.e. at x = 3; similarly, the y -intercept is at y = 2, while the z -intercept is at z = 4. 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 y z as its graph? 1. x + y = 1 2. z - y = 1 3. y + z = 1 correct 4. z - x = 1 5. x + z = 1 6. y - 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 thus look at the trace of the surface on the yz -plane as shown in Consequently, the linear equation is y + z = 1 . keywords: 3Dgraph, linear equation, trace, plane
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Miller, Kierste – Homework 7 – Due: Oct 11 2007, 3:00 am – Inst: JEGilbert 3 004 (part 1 of 1) 10 points Find parametric equations for the line pass- ing through the point P (1 , - 4 , 3) and parallel to the vector h 4 , 1 , - 4 i . 1. x = 1 - 4 t, y = 4 - t, z = 3 - 4 t 2. x = - 1 + 4 t, y = 4 + t, z = - 3 - 4 t 3. x = 4 + t, y = 1 + 4 t, z = 4 - 3 t 4. x = 4 - t, y = - 1 + 4 t, z = - 4 + 3 t 5. x = 1 + 4 t, y = - 4 + t, z = 3 - 4 t correct 6. x = 4 + t, y = 1 - 4 t, z = - 4 + 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 + t v , a
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HW_7_Key - Miller, Kierste Homework 7 Due: Oct 11 2007,...

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