Explain how complex numbers combine algebraically and graphically (solely using the graph, meaning just graphing the result of the algebraic computation is not sufficient) under the following operations:
I understand the first part but the graph part of not be sufficient has stumped
You plot those two complex numbers on the graph. Then create a parallelogram with dotted lines. The resultant vector is the diagonal of the parallelogram from the origin to the vertex. It will be 5 - i, which is just the addition of the real and imaginary components of the two complex numbers. Draw it out and you'll see what I mean.
The division part is a little more complicated. You have to divide the magnitudes of each complex number and subtract their angles.
Using the same example:
magnitude of 3+2i = sqrt(3^2 + 2^2) = sqrt(13)
magnitude of 2-3i = sqrt(2^2 + (-3)^2) = sqrt(13)
The resultant magnitude is sqrt(13) / sqrt(13) = 1
The angle of 3+2i is found using:
tanx = 2/3
x = tan^-1(2/3) = 33.7 degrees
The angle of 2-3i is
x = tan^-1(-3/2) = -56.3 degrees
So, the final vector is 1 with angle of (33.7 -(-56.3) = 90 degrees
Apr 17th, 2015
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