Complex numbers on the Argand diagram Problem set A Worked solutions

The diagram shows the points that represent the complex numbers \(z\) and \(w\).

1. (a) Write down \(z\) and \(w\) as complex numbers.

   • \(z\) is the point \((1, 4)\) and \(w\) is the point \((5, -2)\).

2. (b) Find \(z + w\) and mark it on a copy of the diagram, using the parallelogram rule.

   • \(z + w = (1 + 4 i) + (5 - 2 i) = 6 + 2 i\)
   • Complete the parallelogram with corners \(0\), \(z\), \(w\). The fourth corner is the point \((6, 2)\).

The diagram shows the points that represent \(z\) and \(w\).

1. (a) Write down \(-w\) and mark it on a copy of the diagram.

   • \(w = 3 + 2 i \Rightarrow -w = -3 - 2 i\)
   • \(-w\) is \(w\) reflected through the origin (rotated through \(\pi\) about the origin).

2. (b) Find \(z - w\) and mark it on the diagram.

   • \(z - w = z + (-w) = (-1 + 5 i) + (-3 - 2 i) = -4 + 3 i\)
   • Complete the parallelogram with corners \(0\), \(z\), \(-w\). The fourth corner is \((-4, 3)\).

The diagram shows the points that represent \(z\) and \(\alpha\). The dashed line joins them.

1. (a) Write down \(z\) and \(\alpha\) as complex numbers.

   • \(z\) is the point \((2, 5)\). \(\alpha\) is the point \((6, 1)\).

2. (b) Find \(z - \alpha\).

   • \(z - \alpha = (2 + 5 i) - (6 + i) = -4 + 4 i\)

3. (c) The vector from the point \(\alpha\) to the point \(z\) is \(z - \alpha\). Find the distance between the two points, to 3 significant figures.

   • \(|z - \alpha| = \sqrt{(-4)^2 + (4)^2} = \sqrt{32} = 5.66\)

The triangle PQR has vertices P(2, 0), Q(4, 0) and R(4, 3).

1. (a) Find the modulus and argument of \(1 + i\).

   • \(|1 + i| = \sqrt{(1)^2 + (1)^2} = \sqrt{2}\)
   • \(\arg = \frac{\pi}{4}\) (use the quadrant and the reference angle).

2. (b) Every vertex is multiplied by \(1 + i\) to give the triangle P′Q′R′. Find the new vertices.

   • \(P' = (2)(1 + i) = 2 + 2 i\)
   • \(Q' = (4)(1 + i) = 4 + 4 i\)
   • \(R' = (4 + 3 i)(1 + i) = 1 + 7 i\)

3. (c) Describe the transformation that takes the first triangle to the second.

   • Multiplying by a complex number rotates a point by its argument and scales its distance from the origin by its modulus.
   • So the triangle is rotated through \(\frac{\pi}{4}\) anticlockwise about the origin and enlarged by scale factor \(\sqrt{2}\).

\(OAB\) is an equilateral triangle. \(O\) is the origin and \(A\) is the point \((6, 2)\).

1. (a) Write the rotation of \(A\) through \(\frac{\pi}{3}\) anticlockwise about \(O\) as a multiplication, and find the third vertex \(B\) above the line \(OA\).

   • \(B = (6 + 2 i)\,\mathrm{cis}\left(\tfrac{\pi}{3}\right) = (6 + 2 i)\left(\tfrac{1}{2} + \tfrac{\sqrt{3}}{2}i\right)\)
   • \(= 3 - \sqrt{3} + \left(1 + 3 \sqrt{3}\right) i\)
   • \(B\) is the point \((3 - \sqrt{3}, 1 + 3 \sqrt{3})\).

2. (b) Find the other point \(C\) that makes \(OAC\) equilateral, below the line \(OA\).

   • \(C = (6 + 2 i)\,\mathrm{cis}\left(-\tfrac{\pi}{3}\right) = (6 + 2 i)\left(\tfrac{1}{2} - \tfrac{\sqrt{3}}{2}i\right) = \sqrt{3} + 3 + \left(1 - 3 \sqrt{3}\right) i\)
   • \(C\) is the point \((\sqrt{3} + 3, 1 - 3 \sqrt{3})\).

The diagram shows the points that represent \(z\) and \(w\). Give each argument in the range \(-\pi < \theta \le \pi\).

1. (a) Write down \(z\) and \(w\), and find their moduli and arguments.

   • \(z = -4 + 4i\) is in the second quadrant: \(|z| = \sqrt{16 + 16} = 4\sqrt{2}\), \(\arg z = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\).
   • \(w = 2i\) is on the positive imaginary axis: \(|w| = 2\), \(\arg w = \frac{\pi}{2}\).

2. (b) Find the modulus and argument of \(\dfrac{z}{w}\).

   • \(\left|\frac{z}{w}\right| = \frac{4\sqrt{2}}{2} = 2\sqrt{2}\)
   • \(\arg\frac{z}{w} = \frac{3\pi}{4} - \frac{\pi}{2} = \frac{\pi}{4}\)

3. (c) Describe the transformation that takes the point \(z\) to the point \(\dfrac{z}{w}\), and write \(\dfrac{z}{w}\) in the form \(x + iy\).

   • Dividing by \(w\) turns the point back through \(\arg w = \frac{\pi}{2}\) (clockwise) and divides its distance from the origin by \(|w| = 2\).
   • \(\frac{z}{w} = 2\sqrt{2}\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right) = 2 + 2i\).

The point \(z = 6 + 5i\) is turned through \(\frac{\pi}{2}\) clockwise about the point \(\alpha = 1 + 2i\), shown on the diagram.

1. (a) Find \(z - \alpha\).

   • \(z - \alpha = (6 + 5i) - (1 + 2i) = 5 + 3i\)

2. (b) Multiply \(z - \alpha\) by \(\mathrm{cis}\left(-\frac{\pi}{2}\right) = -i\).

   • \((5 + 3i)(-i) = 3 - 5i\), using \(i^2 = -1\).

3. (c) Hence find the image \(z'\) of \(z\).

   • \(z' = \alpha + (3 - 5i) = 4 - 3i\).
   • Check: \(|z' - \alpha| = |3 - 5i| = |z - \alpha|\).

The point \(z\) is mapped to the point \(w\) by \(w = -2iz + 4\).

1. (a) Find \(w\) when \(z = 1 + 2i\).

   • \(w = -2i(1 + 2i) + 4 = -2i - 4i^2 + 4 = -2i + 4 + 4 = 8 - 2i\)

2. (b) Describe the two transformations that map the point \(z\) onto the point \(w\), in the order they are applied.

   • \(-2i\) has modulus 2 and argument \(-\frac{\pi}{2}\), so multiplying by it rotates \(\frac{\pi}{2}\) clockwise and enlarges by a scale factor 2 about the origin.
   • Adding 4 then translates every point 4 units to the right.

3. (c) Find \(z\) when \(w = 6 + 4i\).

   • \(w - 4 = 2 + 4i\)
   • \(z = \dfrac{2 + 4i}{-2i} = \dfrac{(2 + 4i)(i)}{2} = \dfrac{2i - 4}{2} = -2 + i\)

The complex numbers are \(z = 2 + 4i\) and \(w = 6e^{0.4i}\).

1. (a) Find \(|z|\) and \(\arg z\), to 3 significant figures.

   • \(|z| = \sqrt{2^2 + 4^2} = 4.47\)
   • \(\arg z = \tan^{-1}\dfrac{4}{2} = 1.11\) (first quadrant).

2. (b) Find the angle between the line from \(0\) to \(z\) and the line from \(0\) to \(w\).

   • \(\arg w = 0.4\), so the angle between the lines is \(|0.4 - 1.107| = 0.707\) radians.

3. (c) Find the area of the triangle on an Argand diagram with vertices at \(0\), \(z\) and \(w\), to 3 significant figures.

   • \(\text{Area} = \tfrac{1}{2}|z||w|\sin\varphi = \tfrac{1}{2}(4.472)(6)\sin 0.7071 = 8.72\).

Two tugs pull a barge. Each force, in newtons, is written as a complex number: the real part is the force east and the imaginary part is the force north. The forces are \(z = 300 + 100i\) and \(w = 150 - 250i\).

1. (a) Find the resultant, \(z + w\), of the two.

   • \(z + w = (300 + 100 i) + (150 - 250 i) = 450 - 150 i\)
   • Real parts add and imaginary parts add: this is the parallelogram rule.

2. (b) Find the magnitude of the resultant, to 3 significant figures.

   • \(|z + w| = \sqrt{(450)^2 + (-150)^2} = 474\)

3. (c) Find \(z - w\), the vector from \(w\) to \(z\).

   • \(z - w = (300 + 100 i) - (150 - 250 i) = 150 + 350 i\)

A lighthouse is at the point \(\alpha = 3 + 8i\) on a map, where 1 unit is 1 km. A boat is at the point \(z = 11 + 2i\).

1. (a) Find \(z - \alpha\) and say what it represents.

   • \(z - \alpha = (11 + 2 i) - (3 + 8 i) = 8 - 6 i\)
   • It is the vector (displacement) from the lighthouse \(\alpha\) to the boat \(z\).

2. (b) Find how far the boat is from the lighthouse, to 3 significant figures.

   • \(|z - \alpha| = \sqrt{(8)^2 + (-6)^2} = 10\)

A designer draws a logo as a triangle with vertices A(1, 0); B(3, 0); C(1, 2). A program multiplies every vertex by the complex number \(3 i\).

1. (a) Describe the effect on the shape of multiplying every vertex by that number.

   • \(|3i| = 3,\ \arg 3i = \frac{\pi}{2}\)
   • The shape is rotated through \(\frac{\pi}{2}\) anticlockwise about the origin and enlarged by scale factor \(3\).

2. (b) Find the vertices of the new shape.

   • \(A' = (1) \times (3 i) = 3 i\)
   • \(B' = (3) \times (3 i) = 9 i\)
   • \(C' = (1 + 2 i) \times (3 i) = -6 + 3 i\)

A stage turntable has its centre at \(\alpha = 3 + 2i\), where the units are metres on a plan. A loudspeaker stands at \(z = 8 + 5i\). The turntable turns the loudspeaker through 0.7 radians anticlockwise about its centre.

1. (a) Find \(z - \alpha\).

   • \(z - \alpha = (8 + 5i) - (3 + 2i) = 5 + 3i\)

2. (b) Find the new position of the loudspeaker, with the real and imaginary parts to 3 significant figures.

   • \(z' = \alpha + (z - \alpha)\,\mathrm{cis}\,0.7 = (3 + 2i) + (5 + 3i)(\cos 0.7 + i\sin 0.7)\)
   • \(z' = 4.89 + 7.52i\) (radian mode).

3. (c) Find how far the loudspeaker is from the centre of the turntable after the turn, to 3 significant figures.

   • \(|z' - \alpha| = |z - \alpha| = \sqrt{5^2 + 3^2} = \sqrt{34} = 5.83\) m. A turn does not change the distance from the centre.

\(OPQR\) is a square with \(O\) at the origin and \(P\) at the point \(4 + 2i\). \(R\) is above the line \(OP\).

1. (a) Find the point \(R\), by rotating \(P\) through \(\frac{\pi}{2}\) anticlockwise about \(O\).

   • \(\mathrm{cis}\left(\tfrac{\pi}{2}\right) = i\)
   • \(R = i(4 + 2 i) = -2 + 4 i\)

2. (b) Find the point \(Q\).

   • \(Q = P + R = (4 + 2 i) + (-2 + 4 i) = 2 + 6 i\)
   • The corner \(Q\) completes the parallelogram \(OPQR\), which is a square.

\(PQR\) is an equilateral triangle. \(P\) is the point \(p = 2 + i\) and \(Q\) is the point \(q = 6 + 4i\). \(R\) is on the left of the line \(PQ\) as you go from \(P\) to \(Q\). Neither vertex is at the origin, so use \(q - p\).

1. (a) Find \(q - p\), the vector from \(P\) to \(Q\).

   • \(q - p = (6 + 4 i) - (2 + i) = 4 + 3 i\)

2. (b) Rotate this vector through \(\frac{\pi}{3}\) anticlockwise about \(P\) by multiplying it by \(\mathrm{cis}\left(\frac{\pi}{3}\right)\). Hence find the vector from \(P\) to \(R\).

   • \((4 + 3 i)\left(\tfrac{1}{2} + \tfrac{\sqrt{3}}{2}i\right) = 2 - \frac{3 \sqrt{3}}{2} + \left(\frac{3}{2} + 2 \sqrt{3}\right) i\)

3. (c) Find the coordinates of \(R\).

   • \(r = p + (2 - \frac{3 \sqrt{3}}{2} + \left(\frac{3}{2} + 2 \sqrt{3}\right) i) = 4 - \frac{3 \sqrt{3}}{2} + \left(\frac{5}{2} + 2 \sqrt{3}\right) i\)
   • \(R\) is the point \((4 - \frac{3 \sqrt{3}}{2}, \frac{5}{2} + 2 \sqrt{3})\).