Question

For the complex numbers z = 30(cos(2π/3) + isin(2π/3) and w = 6(cos(π/8) + isin(π/8)), which geometric transformation of z on the complex plane describes z/w?

A) Rotation
B) Scaling
C) Reflection
D) Translation

Answer 1

Dividing complex number z by w results in a combination of the geometric transformations of rotation due to the difference in their arguments and scaling due to the division of their moduli.The correct option is A.

Step-by-step explanation:

When dividing one complex number by another, z by w, there are two geometric transformations that occur: a rotation and a scaling (or dilation). This is because the division of complex numbers in trigonometric form involves dividing their moduli (lengths of the vectors) and subtracting their arguments (angles with the positive x-axis).

The complex number z = 30(cos(2π/3) + isin(2π/3)) has a modulus of 30 and an argument of 2π/3. The complex number w = 6(cos(π/8) + isin(π/8)) has a modulus of 6 and an argument of π/8.

Therefore, to find z/w, you divide the modulus of z by the modulus of w (30/6 = 5), which corresponds to a scaling factor of 5. For the argument, you subtract the argument of w from the argument of z (2π/3 - π/8), which corresponds to a rotation. The correct answer to which geometric transformation of z on the complex plane describes z/w is therefore a combination of scaling and rotation.