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1 We want to find the magnitude ra of the standard-form vector a = (8, 1, 6) Let s call the coordinates xa = 8, ya = 1, and za = 6 Using the formula for vector magnitude, we obtain ra = (xa2 + ya2 + za2)1/2 = [82 + ( 1)2 + ( 6)2]1/2 = (64 + 1 + 36)1/2 = 1011/2 When we round this irrational number to three decimal places, we get ra 10050
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2 The vector a originates at ( 2,0,4) and terminates at (0,0,0) in xyz space To find the standard form of this vector (let s call it a), we subtract the originating coordinates from the terminating ones According to that rule, the x coordinate of a is xa = 0 ( 2) = 2 The y coordinate of a is ya = 0 0 = 0 The z coordinate of a is za = 0 4 = 4 Putting these together, we get a = (xa,ya,za) = (2,0, 4) 3 The vector b originates at (2,3,4) and terminates at (6,7,8) Let b be the standard form of this vector We find the terminating coordinates of b by subtracting the starting coordinates of b from its ending coordinates For the x value of b, we get xb = 6 2 = 4 For the y value of b, we get yb = 7 3 = 4 For the z value of b, we get zb = 8 4 = 4 Assembling these coordinates into an ordered triple, we have b = (xb,yb,zb) = (4,4,4) When we multiply b on the left by the scalar 4, we obtain 4b = 4 (xb,yb,zb) = (4xb,4yb,4zb) = [(4 4),(4 4),(4 4)] = (16,16,16) That s the standard form of 4b, so it must also be the standard form of 4b 4 We have the two standard-form vectors a = ( 7, 10,0) and b = (8, 1, 6)
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506 Worked-Out Solutions to Exercises: 1-9
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Let s assign the coordinates and pair them off as follows: xa = 7 and xb = 8 ya = 10 and yb = 1 za = 0 and zb = 6 When we plug these values into the formula for the dot product of two vectors in Cartesian xyz space, we get a b = xaxb + yayb + zazb = 7 8 + ( 10) ( 1) + 0 ( 6) = 56 + 10 + 0 = 46 5 We have the two standard-form vectors a = (2,6,0) and b = (7,4,3) Let s call the coordinates xa = 2, ya = 6, za = 0, xb = 7, yb = 4, and zb = 3 We can use the formula for the cross product of two vectors in xyz space to get a b = [( yazb zayb),(zaxb xazb),(xayb yaxb)] = [(6 3 0 4),(0 7 2 3),(2 4 6 7)] = [(18 0),(0 6),(8 42)] = (18, 6, 34) 6 We can use the formula for the dot product of two vectors, based on their magnitudes and the angle between them In this case, rf = 4 and rg = 7, and the angle between them, expressible as qfg, is 0 because the vectors point in the same direction Therefore f g = rf rg cos qfg = 4 7 cos 0 = 4 7 1 = 28 7 As in the previous solution, we can use the formula for the dot product of two vectors, based on their magnitudes and the angle between them Here, rf = 4 and rg = 7, and the angle between them, qfg, is p because the vectors point in opposite directions Therefore f g = rfrg cos qfg = 4 7 cos p = 4 7 ( 1) = 28 8 Once again, let s use the formula for the dot product of two vectors, based on their magnitudes and the angle between them The magnitudes are rf = 4 and rg = 7 We see the angle qfg, going counterclockwise from f to g, as p /2 Then f g = rfrg cos qfg = 4 7 cos (p /2) =4 7 0=0
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