Complex Numbers: simultaneous equations and rotation by \(\mathrm{i}\) — NJC 2025 H2 Math Prelim P1
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Question
Do not use a calculator in answering this question.
(i) Find the values of \(z\) and \(w\) that satisfy the equations \((1+\mathrm{i})z+2w=-2+4\mathrm{i}\) and \(3z-w=4+2\mathrm{i}\), expressing your answers in the form \(c+d\mathrm{i}\), where \(c,d\in\mathbf{R}\).
(ii) Points \(W\) and \(Z\) represent \(w\) and \(z\) found in part (i). Find \(\dfrac{w}{z}\) in the form \(p+q\mathrm{i}\), where \(p,q\in\mathbf{R}\). Hence, state the transformation that maps line segment \(OZ\) onto line segment \(OW\).
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(i) Solve the simultaneous equations
Scale the second equation so the \(w\)-terms cancel when added to the first, then isolate \(z\).
\[ (1+\mathrm{i})z + 2w = -2+4\mathrm{i} \quad \ldots (1) \] \[ 3z - w = 4+2\mathrm{i} \quad \ldots (2) \]Multiply (2) by 2: \(\;6z - 2w = 8+4\mathrm{i}\;\) … (3). Adding (1) and (3):
\[ \begin{aligned} (6+1+\mathrm{i})z &= -2+4\mathrm{i}+8+4\mathrm{i} \\ (7+\mathrm{i})z &= 6+8\mathrm{i} \\ z &= \frac{6+8\mathrm{i}}{7+\mathrm{i}} \times \frac{7-\mathrm{i}}{7-\mathrm{i}} \\ &= \frac{42-6\mathrm{i}+56\mathrm{i}+8}{50} \\ &= \frac{50+50\mathrm{i}}{50} \\ &= 1+\mathrm{i} \end{aligned} \]From (2):
\[ \begin{aligned} w &= 3z - 4 - 2\mathrm{i} \\ &= 3(1+\mathrm{i}) - 4 - 2\mathrm{i} \\ &= -1+\mathrm{i} \end{aligned} \]So \(z = 1+\mathrm{i}\) and \(w = -1+\mathrm{i}\).
(ii) Find \(w/z\) and interpret geometrically
Divide by multiplying with the conjugate of the denominator, then read off the rotation that multiplication by \(\mathrm{i}\) represents.
\[ \begin{aligned} \frac{w}{z} &= \frac{-1+\mathrm{i}}{1+\mathrm{i}} \times \frac{1-\mathrm{i}}{1-\mathrm{i}} \\ &= \frac{-1+\mathrm{i}+\mathrm{i}+1}{1^2+1^2} \\ &= \frac{2\mathrm{i}}{2} = \mathrm{i} \end{aligned} \]Since \(w = \mathrm{i} z\), multiplying by \(\mathrm{i}\) corresponds to a rotation of \(\tfrac{\pi}{2}\) anti-clockwise about the origin. Thus \(OW\) is obtained by rotating line segment \(OZ\) by \(\dfrac{\pi}{2}\) anti-clockwise about the origin.