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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)

5

each request with a short paragraph addressing the executive s concern. The key manager on the customer side did the same. Eventually, we won over the doubters and the project moved ahead again. Someone asked how I was able to prepare such clear, concise responses to the questions and challenges I received, and do it so quickly. I replied that I had not done it quickly at all. The customer manager and I had anticipated these issues and answered them in the project overview. Whenever I received an e-mail, I went to the overview, clipped out the paragraph that answered the question, and sent the reply. The time spent early on was well spent. The well-written, thorough overview that the executives would not read got the project through a crisis, preventing con ict, increasing executive commitment, and saving a great deal of time.

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)

496 Worked-Out Solutions to Exercises: 1-9

Once we know what we are going to make, we can plan how we are going to make it The project overview de nes our starting point and our goal, and our next step is to make a detailed work plan The detailed work plan, called a work breakdown structure (WBS), is essential because every other project planning document is based on it From the WBS, we will generate our schedule, our budget, our quality plan, our risk plan, and our procurement plan Therefore, we need to create the WBS very carefully, and ensure that we have not missed any steps This is not as di cult as it sounds In 7 we provide a seven-step plan to creating and validating a WBS Each line in the WBS will be a single task that will produce a deliverable.