In this problem we investigate the effect of the mapping w = az, where a is a complex cons...

In this problem we investigate the effect of the mapping w = az, where a is a complex constant and a ≠ 0, on angles between rays emanating from the origin.

(a) Let C be a ray in the complex plane emanating from the origin. Use parametrizations to show that the image C′ of C under w = az is also a ray emanating from the origin.

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(b) Consider two rays C1 and C2 emanating from the origin such that C1 contains the point z1 = a1+ib1 and C2 contains the point z2 = a2+ib2. In multivariable calculus, you saw that the angle θ between the rays C1 and C2 (which is the same as the angle between the position vectors (a1, b1) and (a2, b2)) is given by:

 (1)

Let  and  be the images of C1 and C2 under w = az. Use part (a) and (14) to show that the angle between  and  s the same as the angle between C1 and C2.