Complex Transformation and Geometry Analysis

00:00 8 marks

Consider the complex number \(z = 2 + 3i\). Apply the following transformations:

(a) Reflect \(z\) across the imaginary axis.
(b) Rotate the resulting complex number by 90° counterclockwise about the origin.
(c) Determine the final position of the point in the complex plane.
(d) Justify geometrically why these transformations yield the final result.

Part a

3 minutes 2 marks

Question

Given the complex number \(z = 2 + 3i\), find its reflection across the imaginary axis.

Hint Hint

Hint

Reflection across the imaginary axis involves changing the sign of the real part.

Answer Answer
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Complete Answer

\( -2 + 3i \)

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Part b

3 minutes 2 marks

Question

Rotate the reflected complex number from part (a) by 90° counterclockwise about the origin.

Hint Hint

Hint

Recall that a 90° counterclockwise rotation about the origin is equivalent to multiplying by \(i\).

Answer Answer
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Explanation Explanation
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Answer Answer

Complete Answer

\( -3 – 2i \)

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Part c

3 minutes 2 marks

Question

State the final position of the point in the complex plane after performing both transformations in sequence.

Hint Hint

Hint

The final position is the complex number obtained after the rotation in part (b).

Answer Answer
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Explanation Explanation
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Answer Answer

Complete Answer

\( -3 – 2i \)

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Part d

3 minutes 2 marks

Question

Geometrically justify why these transformations result in the final position found in part (c), using properties of the Argand diagram.

Hint Hint

Hint

Consider the effect of reflection on the real and imaginary axes and how rotation affects the coordinates in the complex plane.

Answer Answer
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Complete Answer

Reflection across the imaginary axis maps a point \((x, y)\) to \((-x, y)\). A 90° counterclockwise rotation about the origin maps a point \((x, y)\) to \((-y, x)\). Applying these sequentially to \((2, 3)\): Reflection gives \((-2, 3)\), and rotation of \((-2, 3)\) gives \((-3, -2)\), which corresponds to \(-3 – 2i\).

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