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You re on your own That s why you get extra credit! Here s a hint: The work is rather tedious, but it s straightforward
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This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App A The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 Find the magnitude ra of the standard-form vector a = (8, 1, 6) in Cartesian xyz space Assume the values given are exact Using a calculator, round off the answer to three decimal places 2 Imagine a nonstandard vector a that originates at ( 2,0,4) and terminates at the origin Convert a to standard form 3 What s the standard form of the product 4b , where b originates at (2,3,4) and terminates at (6,7,8) Here s a hint: Convert b to its standard form b first, and then multiply that vector by 4
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4 Consider the two standard-form vectors a = ( 7, 10,0) and b = (8, 1, 6) in xyz space What is their dot product 5 Consider the two standard-form vectors a = (2,6,0) and b = (7,4,3) in xyz space What is their cross product 6 Imagine two standard-form vectors f and g that point in the same direction in threespace Suppose that the magnitude rf of f is equal to 4, and the magnitude rg of g is equal to 7 What is f g 7 Imagine two standard-form vectors f and g that point in opposite directions in threespace Suppose that the magnitude rf of vector f is equal to 4, and the magnitude rg of vector g is equal to 7 What is f g 8 Imagine two standard-form vectors f and g that are perpendicular to each other in three-space, and we re looking at them from a point of view such that we see the angle going counterclockwise from f to g as p /2 Suppose that the magnitude rf of vector f is equal to 4, and the magnitude rg of vector g is equal to 7 What is f g What is g f 9 Imagine two standard-form vectors f and g that are perpendicular to each other in three-space, and we re looking at them from a point of view such that we see the angle going counterclockwise from f to g as p /2 Suppose that the magnitude rf of vector f is equal to 4, and the magnitude rg of vector g is equal to 7 What is f g What is g f 10 Consider two standard-form vectors a and b that both lie in the xy plane within Cartesian xyz space Suppose that a = (2,0,0), so it points along the +x axis Suppose that b has magnitude 2 and rotates counterclockwise in the xy plane, starting at (2,0,0), then going around through (0,2,0), ( 2,0,0), and (0, 2,0), finally ending up back at (2,0,0) Now imagine that we watch all this activity from somewhere high above the xy plane, near the +z axis Describe what happens to the cross product vector a b as vector b goes through a complete counterclockwise rotation What will we see if b keeps rotating counterclockwise indefinitely
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CHAPTER
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We can define the locations of points in three dimensions by methods other than the Cartesian system In this chapter, we ll learn about the two most common alternative coordinate schemes for three-space
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