practice testMultiple Choice

Practice Test

Question 5 • Math 4

For two complex numbers z and w , which diagram shows the geometric representation of z w ?

Answer

Image with description A graph has a horizontal axis labeled real and a vertical axis labeled imaginary. Three rays originate from the origin, ordered counter-clockwise from lowest to highest angle. The lowest ray extends to a point labeled w, with a length labeled r 2 and an angle from the horizontal axis labeled theta 2. The middle ray extends to a point labeled z, with a length labeled r 1 and an angle from the horizontal axis labeled theta 1. The highest ray extends to a point labeled z w, with a length labeled r 1 r 2 and an overall angle from the horizontal axis labeled theta 1 + theta 2.

Image with description A graph has a horizontal axis labeled real and a vertical axis labeled imaginary. Three rays originate from the origin, ordered counter-clockwise from lowest to highest angle. The lowest ray extends to a point labeled w, with a length labeled r 2 and an angle from the horizontal axis labeled theta 2. The middle ray extends to a point labeled z w, with a length labeled r 1 r 2 and an angle from the horizontal axis labeled theta 1 minus theta 2. The highest ray extends to a point labeled z, with a length labeled r 1 and an angle from the horizontal axis labeled theta 1.

Image with description A graph has a horizontal axis labeled real and a vertical axis labeled imaginary. Three rays originate from the origin, ordered counter-clockwise from lowest to highest angle. The lowest ray extends to a point labeled w, with a length labeled r 2 and an angle from the horizontal axis labeled theta 2. The middle ray extends to a point labeled z, with a length labeled r 1 and an angle from the horizontal axis labeled theta 1. The highest ray extends to a point labeled z w, with a length labeled r 1 + r 2 and an overall angle from the horizontal axis labeled theta 1 + theta 2.

Image with description A graph has a horizontal axis labeled real and a vertical axis labeled imaginary. Three rays originate from the origin, ordered counter-clockwise from lowest to highest angle. The lowest ray extends to a point labeled w, with a length labeled r 2 and an angle from the horizontal axis labeled theta 2. The middle ray extends to a point labeled z w, with a length labeled r 1 + r 2 and an angle from the horizontal axis labeled theta 1 minus theta 2. The highest ray extends to a point labeled z, with a length labeled r 1 and an angle from the horizontal axis labeled theta 1.

Explanation

Write the complex numbers in polar form: z = r₁(cos θ₁ + i sin θ₁) and w = r₂(cos θ₂ + i sin θ₂). Multiplying complex numbers multiplies their moduli and adds their arguments, so zw has modulus r₁r₂ and angle θ₁ + θ₂. This matches choice A; the other diagrams either use the wrong modulus or subtract the angles.

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