Vectors, Polar Coordinates, and Complex Numbers: math test, Grade 12 – download the PDF

1 Vector basics / 3 pts

Let \(\mathbf{u}=\langle -4,3\rangle\) and \(\mathbf{v}=\langle 5,12\rangle\). (No calculator.)

1. Find \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\).
2. Compute \(2\mathbf{u}+\mathbf{v}\).
3. Compute \(\mathbf{u}\cdot\mathbf{v}\).

2 Work and angle / 3 pts

A force \(\mathbf{F}=\langle 8,6\rangle\) lb moves a crate through the displacement \(\mathbf{d}=\langle 6,8\rangle\) ft.

1. Compute the work \(W=\mathbf{F}\cdot\mathbf{d}\) with its unit.
2. Find \(\|\mathbf{F}\|\) and \(\|\mathbf{d}\|\).
3. With a calculator, find the angle between \(\mathbf{F}\) and \(\mathbf{d}\) to the nearest tenth of a degree.

3 Polar conversions / 4 pts

1. Convert \(\left(6,\dfrac{11\pi}{6}\right)\) to rectangular coordinates.
2. Convert \((-3,3)\) to polar coordinates with \(r>0\) and \(0\le\theta<2\pi\).
3. Convert \(r=8\sin\theta\) to rectangular form and name the graph.

4 A polar curve / 3 pts

Consider the curve \(r=2-2\sin\theta\).

1. Complete the table: \(r\) when \(\theta=0,\ \dfrac\pi6,\ \dfrac\pi2,\ \dfrac{3\pi}{2}\).
2. Name the type of curve and its axis of symmetry.

5 Complex numbers in polar form / 4 pts

Let \(z_1=2\operatorname{cis}100^\circ\) and \(z_2=5\operatorname{cis}35^\circ\).

1. Find \(z_1z_2\) in polar form and in the form \(a+bi\).
2. Find \(z_1^3\) in polar form and in the form \(a+bi\).

6 Cube roots / 3 pts

Solve \(z^3=-64\) in the complex numbers. Give the three solutions in the form \(a+bi\).