Complex numbers on the Argand diagram Problem set B

The Argand diagram shows two complex numbers, \(z\) and \(w\), as points.

1. (a) Write down \(z\) and \(w\) as complex numbers.
2. (b) Find \(z + w\) and mark it on a copy of the diagram, using the parallelogram rule.

Two complex numbers, \(z\) and \(w\), are plotted on the Argand diagram.

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

Two points on the Argand diagram represent \(z\) and \(\alpha\). A dashed line joins them.

1. (a) Write down \(z\) and \(\alpha\) as complex numbers.
2. (b) Find \(z - \alpha\).
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.

The triangle PQR has vertices P(0, 1), Q(0, 3) and R(2, 3).

1. (a) Find the modulus and argument of \(\sqrt{3} + i\).
2. (b) Every vertex is multiplied by \(\sqrt{3} + i\) to give the triangle P′Q′R′. Find the new vertices.
3. (c) Describe the transformation that takes the first triangle to the second.

\(OAB\) is an equilateral triangle. \(O\) is the origin and \(A\) is the point \((5, 1)\).

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

The points \(z\) and \(w\) are shown on the diagram. Give each argument in the range \(-\pi < \theta \le \pi\).

1. (a) Write down \(z\) and \(w\), and find their moduli and arguments.
2. (b) Find the modulus and argument of \(\dfrac{z}{w}\).
3. (c) Describe the transformation that takes the point \(z\) to the point \(\dfrac{z}{w}\), and write \(\dfrac{z}{w}\) in the form \(x + iy\).

The point \(z = 7 + 5i\) is turned through \(\frac{\pi}{2}\) clockwise about the point \(\alpha = 3 + i\), shown on the diagram.

1. (a) Find \(z - \alpha\).
2. (b) Multiply \(z - \alpha\) by \(\mathrm{cis}\left(-\frac{\pi}{2}\right) = -i\).
3. (c) Hence find the image \(z'\) of \(z\).

The point \(z\) is mapped to the point \(w\) by \(w = -3iz + 5\).

1. (a) Find \(w\) when \(z = 2 + i\).
2. (b) Describe the two transformations that map the point \(z\) onto the point \(w\), in the order they are applied.
3. (c) Find \(z\) when \(w = 2 + 6i\).

The complex numbers are \(z = 5 + 2i\) and \(w = 4e^{1.3i}\).

1. (a) Find \(|z|\) and \(\arg z\), to 3 significant figures.
2. (b) Find the angle between the line from \(0\) to \(z\) and the line from \(0\) to \(w\).
3. (c) Find the area of the triangle on an Argand diagram with vertices at \(0\), \(z\) and \(w\), to 3 significant figures.

Two ropes pull on a post. Each force, in newtons, is written as a complex number: the real part is the force to the right and the imaginary part is the force upwards. The forces are \(z = 120 + 340i\) and \(w = -80 + 60i\).

1. (a) Find the resultant, \(z + w\), of the two.
2. (b) Find the magnitude of the resultant, to 3 significant figures.
3. (c) Find \(z - w\), the vector from \(w\) to \(z\).

A radio mast is at the point \(\alpha = -2 + 5i\) on a map, where 1 unit is 1 km. A walker is at the point \(z = 7 - 3i\).

1. (a) Find \(z - \alpha\) and say what it represents.
2. (b) Find how far the walker is from the radio mast, to 3 significant figures.

A tile pattern uses a triangle with vertices A(2, 0); B(2, 2); C(0, 4). A program multiplies every vertex by the complex number \(-2 i\).

1. (a) Describe the effect on the shape of multiplying every vertex by that number.
2. (b) Find the vertices of the new shape.

A Ferris wheel has its hub at \(\alpha = 6 + 10i\) on a plan, in metres, with the ground along the real axis. A seat is level with the hub at \(z = 16 + 10i\). The wheel turns the seat through 0.9 radians anticlockwise about the hub.

1. (a) Find \(z - \alpha\).
2. (b) Find the new position of the seat, with the real and imaginary parts to 3 significant figures.
3. (c) Find how far the seat is from the hub after the turn.

\(OPQR\) is a square with \(O\) at the origin and \(P\) at the point \(5 + i\). \(R\) is above the line \(OP\).

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

\(PQR\) is an equilateral triangle. \(P\) is the point \(p = -2 + i\) and \(Q\) is the point \(q = 3 + 3i\). \(R\) 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 \(Q\).
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 \(R\).
3. (c) Find the coordinates of \(R\).