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Problem 9 in Chap 6 Each radial division represents 1/5 unit
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Complex number 1/2 + j (31/2/2) Pure real number 1 + j0 =1 x
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Figure A-7
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10 Figure A-7 is a graph of three cube roots of 1 as Cartesian complex vectors Each radial division represents 1/5 unit All three vectors terminate on the unit circle
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1 In Fig 7-7, the x axis goes from left to right, the y axis goes from bottom to top, and the z axis goes from far to near According to the following rules: The origin has x = 0, y = 0, and z = 0 The point P has x = 3, y = 3, and z = 4 The point Q has x = 5, y = 4, and z = 0 The point R has x = 0, y = 0, and z = 6
2 We have P = (3, 3,4) Let s call the coordinates xp = 3, yp = 3, and zp = 4 When we plug these values into the formula for the distance c of a point from the origin, we get c = (xp2 + yp2 + zp2)1/2 = [32 + ( 3)2 + 42]1/2 = (9 + 9 + 16)1/2 = 341/2
502 Worked-Out Solutions to Exercises: 1-9
That s an irrational number When we use a calculator to approximate its value to three decimal places, we get c 5831 3 In this case, Q = ( 5,4,0), so we can say that xq = 5, yq = 4, and zq = 0 When we plug these values into the distance-from-the-origin formula, we get c = (xq2 + yq2 + zq2)1/2 = [( 5)2 + 42 + 02]1/2 = (25 + 16 + 0)1/2 = 411/2 A calculator approximates this irrational number to c 6403 4 This distance can be read straightaway from the graph if we use the z axis as a measuring stick If we want to go through the mathematics, we have R = (0,0,6), so we can assign xr = 0, yr = 0, and zr = 6 The distance formula yields c = (xr2 + yr2 + zr2)1/2 = (02 + 02 + 62)1/2 = 361/2 = 6 This value is exact 5 Line segment L connects points Q and R, where Q = (xq,yq,zq) = ( 5,4,0) and R = (xr, yr, zr) = (0,0,6) Plugging the coordinates into the formula for the distance d between two points in Cartesian three-space, we get d = [(xr xq)2 + ( yr yq)2 + (zr zq)2]1/2 = {[0 ( 5)]2 + (0 4)2 + (6 0)2}1/2 = [52 + ( 4)2 + 62]1/2 = (25 + 16 + 36)1/2 = 771/2 When we use a calculator to round this irrational number off to three decimal places, we get d 8775 6 Line segment M connects points P and R, where P = (xp,yp,zp) = (3, 3,4)
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and R = (xr, yr, zr) = (0,0,6) Plugging the coordinates into the formula for the distance d between P and R gives us d = [(xr xp)2 + ( yr yp)2 + (zr zp)2]1/2 = {(0 3)2 + [0 ( 3)]2 + (6 4)2}1/2 = [( 3)2 + 32 + 22]1/2 = (9 + 9 + 4)1/2 = 221/2 A calculator rounds this value to three decimal places as d 4690 7 Line segment N connects points P and Q, where P = (xp,yp,zp) = (3, 3,4) and Q = (xq,yq,zq) = ( 5,4,0) The distance d between these points is d = [(xq xp)2 + ( yq yp)2 + (zq zp)2]1/2 = {( 5 3)2 + [4 ( 3)]2 + (0 4)2}1/2 = [( 8)2 + 72 + ( 4)2]1/2 = (64 + 49 + 16)1/2 = 1291/2 A calculator rounds this to three decimal places as d 11358 8 We want to find the midpoint of the line segment L connecting the points Q = (xq,yq,zq) = ( 5,4,0) and R = (xr, yr, zr) = (0,0,6) Let s call the midpoint A (for average ) in this situation, because we re already using M as the name of a line segment The midpoint formula tells us that the coordinates of A are (xa,ya,za) = [(xq + xr)/2,( yq + yr)/2,(zq + zr)/2] = {( 5 + 0)/2,(4 + 0)/2,(0 + 6)/2} = ( 5/2,4/2,6/2) = ( 5/2,2,3)
504 Worked-Out Solutions to Exercises: 1-9
9 We want to find the midpoint of the line segment M connecting the points P = (xp,yp,zp) = (3, 3,4) and R = (xr,yr,zr) = (0,0,6) If we again call the midpoint A, our formula tells us that the coordinates of A are (xa,ya,za) = [(xp + xr)/2,( yp + yr)/2,(zp + zr)/2] = [(3 + 0)/2,( 3 + 0)/2,(4 + 6)/2} = (3/2, 3/2,10/2) = (3/2, 3/2,5) 10 We want to identify the midpoint of the line segment N connecting the points P = (xp,yp,zp) = (3, 3,4) and Q = (xq,yq,zq) = ( 5,4,0) Let s call the midpoint A once more Plugging the values into the formula, we obtain the coordinates (xa,ya,za) = [(xp + xq)/2,( yp + yq)/2,(zp + zq)/2] = {[(3 + ( 5)]/2,( 3 + 4)/2,(4 + 0)/2} = ( 2/2,1/2,4/2) = ( 1,1/2,2)
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