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Let s assign the coordinate values qa = p /4, qb = 3p /4, ra = 321/2, and rb = 981/2 The Cartesian polar product of a and b, in that order, is a b = rarb cos (qb qa) = 321/2 981/2 cos (3p /4 p /4) = 3,1361/2 cos (p /2) = 56 0 = 0 The Cartesian dot product of b and a, in that order, is b a = rbra cos (qa qb) = 981/2 321/2 cos (p /4 3p /4) = 3,1361/2 cos ( p /2) = 56 0 = 0 6 Consider two standard-form vectors a and b in Cartesian coordinates, defined by the ordered pairs a = (xa,ya) and b = (xb,yb) By definition, the Cartesian dot product of a and b, in that order, is a b = xaxb + yayb The commutative law for real-number multiplication allows us to reverse the order of both terms in the sum on the right-hand side of this equation, getting a b = xbxa + ybya By definition, the right-hand side of the above equation is the Cartesian dot product of b and a, in that order Therefore a b=b a for any two standard-form Cartesian-plane vectors a and b 7 Suppose we re given two vectors a and b in the polar plane, defined by a = (qa,ra) and b = (qb,rb) The polar dot product of a and b, in that order, is a b = rarb cos (qb qa)
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The commutative law for real-number multiplication allows us to reverse the order of the multiplication on the right-hand side of this equation to obtain a b = rbra cos (qb qa) Now let s look at the difference between the direction angles From pre-algebra, we recall that when we reverse the order of a difference, we get the negative of that difference Using this rule, we can modify the angular difference in the above equation to get a b = rbra cos [ (qa qb)] Basic trigonometry tells us that the cosine of the negative of an angle is the same as the cosine of the angle itself Therefore a b = rbra cos (qa qb) By definition, the right-hand side of this equation is the polar dot product of b and a, in that order, telling us that a b=b a for any two vectors a and b in the polar plane 8 Let s do the Cartesian proof first We have a positive scalar k+ along with two standardform vectors a and b in the xy plane Suppose that the coordinates are a = (xa,ya) and b = (xb,yb) When we multiply these vectors individually on the left by k+, we get k+a = (k+xa,k+ya) and k+b = (k+xb,k+yb) The Cartesian dot product of these vectors is k+a k+b = (k+xak+xb + k+yak+yb) = (k+2xaxb + k+2yayb) = k+2(xaxb + yayb) = k+2(a b) Now let s work through the polar case Suppose that the coordinates of a and b are a = (qa,ra)
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and b = (qb,rb) When we multiply the individual vectors on the left by the positive scalar k+ and expand the results into ordered pairs, we get k+a = (qa,k+ra) and k+b = (qb,k+rb) Does this step confuse you If so, remember that because the scalar k+ is positive, multiplying any polar vector by k+ doesn t change the vector direction It only affects the magnitude, making it k+ times as large When we take the polar dot product of these new vectors, we get k+a k+b = k+rak+rb cos (qb qa) = k+2rarb cos (qb qa) = k+2(a b) 9 We want to find the cross product a b of the polar vectors a = (p /3,4) and b = (3p /2,1) The coordinate values are qa = p /3, qb = 3p /2, ra = 4, and rb = 1 Before we begin our calculations, we should note that qb qa = 3p /2 p /3 = 7p /6 That s larger than p, so a b points straight away from us as we look down on the polar plane To find the magnitude ra b, we use the formula for cases where p < qb qa < 2p That gives us ra b = rarb sin (2p + qa qb) = 4 1 sin (5p /6) = 4 1 1/2 = 2 so we know that the magnitude of a b is 2 10 We ve been told to find the cross product of the polar vectors a = (p,8)
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and b = (7p /6,5) The coordinate values are qa = p, qb = 7p /6, ra = 8, and rb = 5 In this situation, we have qb qa = 7p /6 p = p /6 That s smaller than p, so the cross product vector points directly toward us as we look down on the polar plane and imagine going counterclockwise from a to b To find the magnitude ra b, we use the formula for situations in which 0 < qb qa < p, getting ra b = rarb sin (qb qa) = 8 5 sin (p /6) = 8 5 1/2 = 20 so we know that the magnitude of a b is 20
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