Write the trigonometric forms of the complex numbers.

$$\frac{4i}{-4-2i}$$

Perform the indicated operation using the trigonometric forms.

$$\frac{5}{2+3i}$$

Find the component form of the sum of $\bf{u}$ and $\bf{v}$ with the given magnitudes and direction angles $\bf{\theta_u}$ and $\bf{\theta_v}$.

$$\underline{\text{Magnitude}}\ \ \ \ \ \ \ \ \underline{\text{Angle}}$$

$$\lVert {\bf{u}}\rVert= 20\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\bf{\theta_u}}=45^\circ$$

$$\lVert {\bf{v}}\rVert= 50\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\bf{\theta_v}}=180^\circ$$

Find the magnitude and the direction angle of the vector $\bf{v}$.

$${\bf{v}}=-3{\bf{i}}-3\bf{j}$$

Write the trigonometric forms of the complex numbers.

$$\frac{5}{2-3i}$$

Find the component form of the sum of $\bf{u}$ and $\bf{v}$ with the given magnitudes and direction angles $\bf{\theta_u}$ and $\bf{\theta_v}$.

$$\underline{\text{Magnitude}}\ \ \ \ \ \ \ \ \underline{\text{Angle}}$$

$$\lVert {\bf{u}}\rVert= 4\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\bf{\theta_u}}=60^\circ$$

$$\lVert {\bf{v}}\rVert= 4\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\bf{\theta_v}}=90^\circ$$

Find the magnitude and the direction angle of the vector $\bf{v}$.

$${\bf{v}}=-4{\bf{i}}+7\bf{j}$$

Prove the following Properties of the Dot Product.

Prove that $4(\bf{u\cdot v})=\lVert \bf{u+v}\rVert^2 -\lVert \bf{u-v}\rVert^2$

Perform the indicated operation using the standard forms, and check your result with that of part (b).

$$\frac{1+\sqrt3+i}{6-3i}$$

A triangle is to be formed by drawing a line segment of length $a$ from $(4,3)$ to the positive $x$-axis. For what value(s) of $a$ can you form no triangles? Explain your reasoning.

Perform the indicated operation using the trigonometric forms.

$$\frac{1+\sqrt3+i}{6-3i}$$

Find the magnitude and the direction angle of the vector $\bf{v}$.

$${\bf{v}}=5{\bf{i}}+4\bf{j}$$

Write the trigonometric forms of the complex numbers.

$$\frac{1+\sqrt3i}{6-3i}$$

A triangle is to be formed by drawing a line segment of length $a$ from $(4,3)$ to the positive $x$-axis. For what value(s) of $a$ can you form two triangles? Explain your reasoning.