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When you come across a messy ordered triple like this, you might ask, Is there any way to make it look simpler Sometimes there is In this case, there isn t You can get rid of the grouping symbols if you re willing to use a calculator to approximate the values But even if you do that, you ll have to remember that in spherical coordinates, the first two values represent angles in radians, and the third value represents a linear distance
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Suppose we re given the coordinates of a point P in spherical three-space as P = (q,f,r) = (3p /4,p /4,31/2) Find the coordinates of P in cylindrical coordinates
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We haven t learned any formulas for direct conversion between spherical and cylindrical coordinates, so we must convert to Cartesian coordinates first, and then to cylindrical coordinates from there The Cartesian x value is x = r sin f cos q = 31/2 sin (p /4) cos (3p /4) = 31/2 21/2/2 ( 21/2/2) = 31/2/2
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The Cartesian y value is y = r sin f sin q = 31/2 sin (p /4) sin (3p /4) = 31/2 21/2/2 21/2/2 = 31/2/2 The Cartesian z value is z = r cos f = 31/2 cos (p /4) = 31/2 21/2/2 = 61/2/2 Our Cartesian ordered triple is therefore P = (x,y,z) = ( 31/2/2,31/2/2,61/2/2) Now let s convert these coordinates to their cylindrical counterparts We have x = 31/2/2 and y = 31/2/2 To find the cylindrical direction angle q, we use the formula q = p + Arctan ( y /x) because x < 0 and y > 0 When we plug in the values for x and y, we get q = p + Arctan [(31/2/2) / ( 31/2/2)] = p + Arctan ( 1) = p + ( p /4) = 3p /4 This is the same as the horizontal direction angle in the original set of spherical coordinates, as we should expect (If things hadn t come out that way, we d have made a mistake!) When we input the values for x and y to the formula for the cylindrical radius r, we get r = [( 31/2/2)2 + (31/2/2)2]1/2 = (3/4 + 3/4)1/2 = (6/4)1/2 = 61/2/2 We calculated that z = 61/2/2, so the cylindrical height h is h = z = 61/2/2 We ve found that the cylindrical equivalent point is (q,r,h) = (3p /4,61/2/2,61/2/2)
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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 Describe the graphs of the following equations in cylindrical coordinates What would they look like in Cartesian xyz space q=0 r=0 h=0 2 Plot the point (q,r,h) = (3p /4,6,8) in the cylindrical coordinate system 3 Consider the point (q,r,h) = (p /4,0,1) in cylindrical coordinates Find the equivalent of this point in Cartesian xyz space 4 Consider the point ( 4,1,0) in xyz space Find the equivalent of this point in cylindrical three-space First, find the exact coordinates Then, using a calculator, approximate the irrational coordinates to four decimal places 5 In the chapter text, we used the conversion formulas to find that the cylindrical equivalent of (x,y,z) = (1,1,1) is (q,r,h) = (p /4,21/2,1) Convert these coordinates back to Cartesian xyz coordinates to verify that the result we got was correct and unambiguous 6 Describe the graphs of the following equations in spherical coordinates What would they look like in Cartesian xyz space q=0 f=0 r=0 7 Plot the point (q,f,r) = (3p /4,p /4,8) in the spherical coordinate system 8 Consider the point (q,f,r) = (p /4,0,1) in spherical coordinates Find the equivalent of this point in Cartesian xyz space 9 Consider the point ( 4,1,0) in xyz space Find the equivalent of this point in spherical three-space First, find the exact coordinates Then, using a calculator, approximate the irrational coordinates to four decimal places 10 Work the final challenge backward to verify that we did our calculations correctly Consider the point P in cylindrical three-space given by P = (q,r,h) = [3p /4,61/2/2,61/2/2] Find the coordinates of P in Cartesian coordinates, and from there, convert to spherical coordinates
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