The complex plane and the modulus Problem set A One question at a time

The diagram shows the point that represents \(z = 5 + 2i\).

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

   • \(-z = -5 - 2i\), the point \((-5, -2)\)
   • A reflection of z through the origin.

2. (b) Write down \(z^*\) and mark it on the diagram.

   • \(z^* = 5 - 2i\), the point \((5, -2)\)
   • A reflection of z in the real axis.

3. (c) Write down \(-z^*\) and mark it on the diagram. Describe the reflection that takes \(z\) to \(-z^*\).

   • \(-z^* = -5 + 2i\), the point \((-5, 2)\)
   • The imaginary part is unchanged and the real part changes sign: a reflection in the imaginary axis.

The diagram shows two points, P and Q, on an Argand diagram.

1. (a) Write down the complex number represented by P.

   • \(P \text{ is the point } (-12, 16) \text{, so the number is } -12 + 16i\)

2. (b) Find \(|P|\).

   • \(|P| = \sqrt{(-12)^2 + 16^2} = 20\)

3. (c) Q represents \(w\). Find \(|w|\) to 3 significant figures.

   • \(w = -6 - 2i\)
   • \(|w| = \sqrt{(-6)^2 + (-2)^2} = \sqrt{40} = 6.32\)

\(z = -4 + 3i\).

1. (a) Write down \(-z\) and say how its point is related to the point for \(z\).

   • \(-z = 4 - 3i\)
   • Both parts change sign: the point is the reflection of z through the origin.

2. (b) Write down \(z^*\) and say how its point is related to the point for \(z\).

   • \(z^* = -4 - 3i\)
   • Only the imaginary part changes sign: the reflection of z in the real axis.

3. (c) Find \(|z|\), \(|z^*|\) and \(|-z|\).

   • \(|z| = \sqrt{(-4)^2 + 3^2} = \sqrt{25} = 5\)
   • The other two points are reflections of z, so they are the same distance from the origin.

Find each modulus. Give an exact value if it is an integer, and otherwise give 3 significant figures.

1. (a) \(|7 - 24i|\)

   • \(|7 - 24i| = \sqrt{7^2 + (-24)^2} = \sqrt{625} = 25\)

2. (b) \(|-5 - 3i|\)

   • \(|-5 - 3i| = \sqrt{(-5)^2 + (-3)^2} = \sqrt{34} = 5.83\)

3. (c) \(|12i|\)

   • \(|12i| = \sqrt{0^2 + 12^2} = \sqrt{144} = 12\)

Take east as the real axis and north as the imaginary axis, with distances in metres. A drone starts at the origin and flies to the point \(z = 9 - 12i\).

1. (a) Find the distance of the drone from the origin.

   • \(|z| = \sqrt{9^2 + (-12)^2} = 15\)

2. (b) A second drone is at \(z^*\). Find the distance between the two drones.

   • \(z^* = 9 + 12i\)
   • \(z - z^* = (9 - 12i) - (9 + 12i) = -24i\)
   • \(|-24i| = 24\), or: the drones are 12 above and 12 below the real axis.

\(z\) is a complex number with \(|z| = 5\) and \(\text{Re}(z) = -3\).

1. (a) Find the two possible complex numbers \(z\).

   • \(z = -3 + iy\)
   • \(|z| = \sqrt{(-3)^2 + y^2} = 5\)
   • \(9 + y^2 = 25,\quad y^2 = 16,\quad y = \pm 4\)

2. (b) How are the two points related on the Argand diagram?

   • \(-3 + 4i\) and \(-3 - 4i\) have the same real part and opposite imaginary parts.
   • So each is the conjugate of the other: a reflection in the real axis.

The points that represent \(z = 5 + 3i\), \(z^*\), \(-z\) and \(-z^*\) are the corners of a rectangle on an Argand diagram.

1. (a) Find the width and height of the rectangle.

   • \(z = 5 + 3i, \quad z^* = 5 - 3i\)
   • The corners are \((5, 3)\), \((5, -3)\), \((-5, -3)\), \((-5, 3)\).
   • Width \(= 2 \times 5 = 10\), height \(= 2 \times 3 = 6\).

2. (b) Find its perimeter and its area.

   • \(\text{Perimeter} = 2(10 + 6) = 32\)
   • \(\text{Area} = 10 \times 6 = 60\)