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In this section we talk about the roles of the project manager and other people on a project. Please note that there is a di erence between roles and people. One person can have several roles. For example, writing this book is a project, and I am the project manager. I am also the author the person doing the writing. So I have two roles. Sometimes, a single role can be divided between two people. This only works if the people work together well. But there is an excellent buddy system of computer programming where two people write code together. While one is writing, the other is looking for errors or thinking of better ways to write the routine. They can also think through problems together. The buddy method, where two people share one role, reduces errors so much that it actually saves time compared to having just one programmer.

P = ( 5, 2) and Q = (7,3) When we plug xp = 5, yp = 2, xq = 7, and yq = 3 into the midpoint formula, we get (xm,ym) = [(xp + xq)/2,(yp + yq)/2] = [( 5 + 7)/2,( 2 + 3)/2] = (2/2,1/2) = (1,1/2)

It seems reasonable to suppose the midpoint between points P and Q should not depend on whether we go from P to Q or from Q to P We can prove this by showing that for all real numbers xp, yp, xq, and yq, we have [(xp + xq)/2,(yp + yq)/2] = [(xq + xp)/2,(yq + yp)/2] This demonstration is easy, but let s go through it step-by-step to completely follow the logic For the x coordinates, the commutative law of addition tells us that xp + xq = xq + xp Dividing each side by 2 gives us (xp + xq)/2 = (xq + xp)/2 For the y coordinates, the commutative law says that yp + yq = yq + yp

We can have nonstandard vertical angles, although things are simplest if we keep them nonnegative but no larger than p Theoretically, all possible locations in space can be covered if we restrict the vertical angle to the range 0 f p If it s outside this range, such as p < f < 0 or p < f < 2p, we can multiply the radius r by 1, and then add or subtract p to or from f, and we ll end up at the point we want But those are confusing ways to get there! The radius r can be any real number, but things are simplest if we keep it nonnegative If our horizontal and vertical direction angles put us on a ray that goes from the origin through P, then r > 0 If our direction angles put us on a ray that goes from the origin away from P, then r < 0 We have r = 0 if and only if P is at the origin If we find ourselves working with a negative radius, we should reverse the direction by adding or subtracting p to or from both angles, keeping 0 q < 2p and 0 f p Then we can take the absolute value of the negative radius and use it as the radius coordinate

The intercept po is the static pressure at no flow. Differentiating, we obtain t he process gain: (3.6)

Role A de ned set of responsibilities in relation to a project, work environment, or social situation.

An example In the situation of Fig 9-5, the horizontal direction angle q appears to be somewhere between p and 3p /2 The vertical direction angle f appears to be roughly 1 radian We can t be sure of the exact values of these angles, because we don t have any reference lines to compare them with The radius r is positive, but we have no idea how large it is because there are no radial coordinate increments Another example Imagine that we set the horizontal direction angle q equal to a constant in spherical coordinates For example, let s say that we have the equation

q = 7p /5 When we work in polar coordinates and set the direction angle equal to a constant, we get a line passing through the origin In spherical coordinates, the horizontal angle in the reference plane is geometrically identical to the polar direction angle Therefore, if k is any real-number constant, the graph of q=k is a plane that passes through the vertical axis When k = 7p /5, that vertical plane also contains the ray for the direction angle 7p /5, as shown in Fig 9-6