vb.net generate qr code How can we convert a Cartesian complex vector to a polar complex vector in .NET

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How can we convert a Cartesian complex vector to a polar complex vector
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Answer 6-8
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Imagine a complex number a + jb in the Cartesian complex plane, whose vector extends from the origin to the point (a,jb) We can derive the magnitude r of the equivalent polar vector by applying the distance formula to get
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r = (a2 + b2)1/2
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x x=a Ordered pair is (a, jb) representing the complex number a + jb
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Figure 10-4 Illustration for Question and Answer 6-7
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Ordered pair is (q, r) representing the complex number a + jb shown in Fig 10-4
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Figure 10-5 Another illustration for Question and Answer 6-7
Part One
To determine the direction angle q of the polar vector, we modify the polar-coordinate direction-finding 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 6-9
How can we convert a polar complex vector to a Cartesian complex vector
Answer 6-9
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 6-10
What are the two versions of De Moivre s theorem How are they used
Answer 6-10
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 real-number polar magnitudes, and q1 and q2 are real-number 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 real-number polar magnitude and q is the real-number polar angle Also suppose that n is an integer Then cn = rn cos (nq) + j[rn sin (nq)]
7
Question 7-1
How are the axes and variables defined in Cartesian xyz space
Answer 7-1
We construct Cartesian xyz space by placing three real-number 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 10-6 shows two perspective drawings of the typical system Although the point of Figure 10-6 Illustration for
Question and Answer 7-1
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 7-2
What s the difference between Cartesian xyz space and rectangular xyz space
Answer 7-2
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 7-3
What s the pool rule for the relative axis orientation and coordinate values in Cartesian xyz space
Answer 7-3
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 7-4
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