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

1 Magnitude of a vector ★★★

Find the magnitude of each vector.

1. \(\langle 5,12\rangle\)
2. \(\langle -8,6\rangle\)
3. \(\langle 7,-24\rangle\)

2 Direction angles ★★★

Find the magnitude and the direction angle (from \(0^\circ\) to \(360^\circ\)) of each vector.

1. \(\langle 1,\sqrt3\rangle\)
2. \(\langle -2,-2\rangle\)
3. \(\langle 3,-3\sqrt3\rangle\)

3 Combining vectors ★★★

Let \(\mathbf{u}=\langle 4,-1\rangle\) and \(\mathbf{v}=\langle -2,6\rangle\). Compute \(\mathbf{u}+\mathbf{v}\), \(\mathbf{u}-\mathbf{v}\), and \(3\mathbf{u}-2\mathbf{v}\).

4 Computing dot products ★★★

Compute each dot product and say whether the vectors are perpendicular.

1. \(\langle 3,5\rangle\cdot\langle -2,4\rangle\)
2. \(\langle 6,-3\rangle\cdot\langle 1,2\rangle\)
3. \(\langle 2,2\rangle\cdot\langle -1,-1\rangle\)

5 Polar to rectangular ★★★

Convert each polar point to rectangular coordinates.

1. \(\left(8,\dfrac{\pi}{3}\right)\)
2. \((5,\pi)\)
3. \(\left(6,\dfrac{3\pi}{2}\right)\)

6 Rectangular to polar ★★★

Give polar coordinates \((r,\theta)\) with \(r>0\) and \(0\le\theta<2\pi\).

1. \((0,9)\)
2. \((-4,0)\)
3. \((3,3)\)

7 Polar form of complex numbers ★★★

Write each complex number in polar form \(r\operatorname{cis}\theta\) with \(0^\circ\le\theta<360^\circ\).

1. \(2+2i\)
2. \(-6\)
3. \(-3i\)

8 True or false? ★★★

Decide whether each statement is true or false and justify your answer with a counterexample or a short reason.

1. \(\|\mathbf{u}+\mathbf{v}\|=\|\mathbf{u}\|+\|\mathbf{v}\|\) for all vectors \(\mathbf{u},\mathbf{v}\).
2. If \(\mathbf{u}\cdot\mathbf{v}=0\), then \(\mathbf{u}\) or \(\mathbf{v}\) is the zero vector.
3. The polar points \((3,\theta)\) and \((-3,\theta)\) are the same point.

9 Kayak in a current ★★★

A kayaker paddles due north at 5 mph (about 8.0 km/h) while a current flows due east at 3 mph. Take east as the positive \(x\)-direction and north as the positive \(y\)-direction.

1. Write the paddling and current vectors, then the resultant velocity.
2. Find the kayaker’s actual speed in mph and in km/h.
3. Find the direction of travel as an angle east of north.

10 Pulling a sled ★★★

A rope pulls a sled with a constant force of 50 lb directed \(30^\circ\) above the horizontal. The sled moves 20 ft horizontally. Find the work done, in foot-pounds.

11 Angle between two vectors ★★★

Find the angle between \(\mathbf{u}=\langle 2,1\rangle\) and \(\mathbf{v}=\langle 1,3\rangle\).

12 Polar equation to rectangular ★★★

Convert to rectangular form and identify the graph.

1. \(r=6\cos\theta\)
2. \(r\sin\theta=4\)

13 Rectangular equation to polar ★★★

Write each equation in polar form.

1. \(x^2+y^2=25\)
2. \(y=\sqrt3\,x\)
3. \(y=-3\)

14 Products and quotients ★★★

Let \(z_1=4\operatorname{cis}50^\circ\) and \(z_2=2\operatorname{cis}20^\circ\). Find \(z_1z_2\) and \(\dfrac{z_1}{z_2}\) in polar form, then write \(\dfrac{z_1}{z_2}\) as \(a+bi\).

15 Powers with De Moivre ★★★

Use De Moivre’s Theorem to evaluate.

1. \((\cos20^\circ+i\sin20^\circ)^9\)
2. \((2\operatorname{cis}15^\circ)^4\), in the form \(a+bi\)

16 Cardioid and roses ★★★

(a) Complete a table of values for \(r=2+2\cos\theta\) at \(\theta=0,\ \dfrac\pi3,\ \dfrac\pi2,\ \dfrac{2\pi}{3},\ \pi\), and state the symmetry of the graph.

(b) How many petals does \(r=5\cos3\theta\) have, and how long is each? (c) How many petals does \(r=4\sin2\theta\) have?

17 Where two curves meet ★★★

Find the points where the circles \(r=3\) and \(r=6\cos\theta\) intersect. Give the answers in polar and rectangular coordinates.

18 Projection of a vector ★★★

Let \(\mathbf{u}=\langle 6,2\rangle\) and \(\mathbf{v}=\langle 4,-3\rangle\). Find the scalar component of \(\mathbf{u}\) along \(\mathbf{v}\), the projection \(\operatorname{proj}_{\mathbf{v}}\mathbf{u}\), and verify that \(\mathbf{u}-\operatorname{proj}_{\mathbf{v}}\mathbf{u}\) is perpendicular to \(\mathbf{v}\).

19 A triangle from vectors ★★★

Triangle \(ABC\) has vertices \(A(1,1)\), \(B(4,5)\), and \(C(8,2)\). Use vectors to show that the triangle has a right angle at \(B\) and is isosceles, then find its area.

20 A tenth power ★★★

Compute \((-1+i)^{10}\) with De Moivre’s Theorem, then check your result by squaring first.

21 Sixth roots of 64 ★★★

Find all complex solutions of \(z^6=64\). Write them in the form \(a+bi\), and identify the six points \(z_0,\dots,z_5\) of the figure (labeled counterclockwise from the positive real axis) with these values.

What is the sum of the six roots?

22 Fourth roots ★★★

Find the four fourth roots of \(-8+8\sqrt3\,i\) in the form \(a+bi\), and check one of them by raising it to the fourth power.

23 Triple-angle formulas ★★★

Expand \((\cos\theta+i\sin\theta)^3\) with the binomial theorem and compare with De Moivre’s Theorem to prove \(\cos3\theta=4\cos^3\theta-3\cos\theta\). What formula for \(\sin3\theta\) do you get?