Unit 9 test: Vectors and Polar Form

Find the component form of the vector from $P(4, -2)$ to $Q(-2, 5)$. Enter it as $(a, b)$.

Find the magnitude of $\mathbf{v} = 7\mathbf{i} - 24\mathbf{j}$.

A vector has magnitude $12$ and direction angle $120^\circ$. Find its component form. Enter it as $(a, b)$.

Find $\langle 5, -2 \rangle \cdot \langle 1, 4 \rangle$.

Find the angle between $\mathbf{u} = \langle 1, 2 \rangle$ and $\mathbf{v} = \langle 3, -1 \rangle$ in degrees, rounded to the nearest tenth.

Find $k$ so that $\langle 5, 2 \rangle$ and $\langle 4, k \rangle$ are orthogonal.

Convert the polar point $(6, \tfrac{2\pi}{3})$ to rectangular coordinates.

Write the rectangular point $(\sqrt{3}, -1)$ in polar form with $r > 0$ and $0 \le \theta < 2\pi$. Give $\theta$ in radians.

Which rectangular equation has the same graph as $r = 4\cos\theta$?

How many petals does the rose $r = 7\cos(6\theta)$ have?

What kind of graph is $r = 3 + 3\sin\theta$?

Write $z = -4 - 4i$ in trigonometric form $r(\cos\theta + i\sin\theta)$ and enter $(r, \theta)$, with $\theta$ in degrees and $0^\circ \le \theta < 360^\circ$.

Let $z_1 = 10(\cos 130^\circ + i\sin 130^\circ)$ and $z_2 = 2(\cos 10^\circ + i\sin 10^\circ)$. Find the real part of $\dfrac{z_1}{z_2}$.

Find $(1 + i)^4$. The answer is a real number.

The cube roots of $8(\cos 120^\circ + i\sin 120^\circ)$ all have modulus $2$. List their arguments in degrees, with $0^\circ \le \theta < 360^\circ$, separated by commas.