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Complex Rotations

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Solve 1-4 in the attachment.

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Solution Summary

We solve several problems pertaining to complex multiplication and rotation in the complex plane.

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4.7.1. Say v = v_x + i v_y and w = w_x + i w_y. By the definition of absolute value of complex numbers, we have |v - w| = |(v_x - w_x) + i(v_y - w_y)| = sqrt((v_x - w_x)^2 + (v_y + w_y)^2), which we know is the Euclidean distance from the point v = (v_x, v_y) to the point w = (w_x, w_y).

4.7.2. Let d(z1, z2) denote the distance between two complex numbers, z1 and z2. By problem 1, we see that

d(z1, z2) = |z1 - z2|.

Thus we ...

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