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vb.net generate qr code How can we convert a Cartesian complex vector to a polar complex vector in .NET
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To determine the direction angle q of the polar vector, we modify the polarcoordinate directionfinding system Here s what happens: When a = 0 and jb = j0, we have q = 0 by default When a > 0 and jb = j0, we have q = 0 When a > 0 and jb > j0, we have q = Arctan (b /a) When a = 0 and jb > j0, we have q = p /2 When a < 0 and jb > j0, we have q = p + Arctan (b /a) When a < 0 and jb = j0, we have q = p When a < 0 and jb < j0, we have q = p + Arctan (b /a) When a = 0 and jb < j0, we have q = 3p /2 When a > 0 and jb < j0, we have q = 2p + Arctan (b /a) Question 69 How can we convert a polar complex vector to a Cartesian complex vector
Answer 69 Imagine a complex vector (q,r) in the polar complex plane, whose direction angle is q and whose radius is r The Cartesian vector equivalent is (a,jb) = [(r cos q), j(r sin q)] which represents the complex number a + jb = r cos q + j(r sin q) Question 610 What are the two versions of De Moivre s theorem How are they used
Answer 610 The first, and more general, version of De Moivre s theorem involves products and ratios Suppose we have two polar complex numbers c1 and c2, where c1 = r1 cos q1 + j(r1 sin q1) and c2 = r2 cos q2 + j(r2 sin q2) where r1 and r2 are realnumber polar magnitudes, and q1 and q2 are realnumber polar angles in radians Then c1c2 = r1r2 cos (q1 + q2) + j [r1r2 sin (q1 + q2)] and, as long as r2 is nonzero, c1/c2 = (r1/r2) cos (q1 q2) + j [(r1/r2) sin (q1 q2)] Review Questions and Answers
The second version of De Moivre s theorem involves integer powers Suppose that c is a complex number, where c = r cos q + j(r sin q) where r is the realnumber polar magnitude and q is the realnumber polar angle Also suppose that n is an integer Then cn = rn cos (nq) + j[rn sin (nq)] 7
Question 71 How are the axes and variables defined in Cartesian xyz space
Answer 71 We construct Cartesian xyz space by placing three realnumber lines so that they all intersect at their zero points, and they re all mutually perpendicular One number line represents the variable x, another represents the variable y, and the third represents the variable z Figure 106 shows two perspective drawings of the typical system Although the point of Figure 106 Illustration for Question and Answer 71 Part One
view differs between illustrations A and B, the relative axis orientation is the same in both cases When we graph relations and functions having two independent variables in Cartesian xyz space, x and y are usually the independent variables, and z is usually the dependent variable Question 72 What s the difference between Cartesian xyz space and rectangular xyz space
Answer 72 In Cartesian xyz space, the axes are all linear, and they re all graduated in increments of the same size In rectangular xyz space, the divisions can differ in size between the axes, although each axis must be linear along its entire length Question 73 What s the pool rule for the relative axis orientation and coordinate values in Cartesian xyz space Answer 73 We can imagine that the origin of the coordinate grid rests on the surface of a swimming pool We orient the positive x axis horizontally along the pool surface, pointing due east Once we ve done that, the coordinate values can be generalized as follows: Positive values of x are east of the origin Negative values of x are west of the origin Positive values of y are north of the origin Negative values of y are south of the origin Positive values of z are up in the air Negative values of z are under the water Question 74

