Chapter 7

Use the given vectors to find \(\mathbf{v} \cdot \mathbf{w}\) and \(\mathbf{v} \cdot \mathbf{v}\). $$ \mathbf{v}=3 \mathbf{i}+\mathbf{j}, \quad \mathbf{w}=\mathbf{i}+3 \mathbf{j} $$

In Exercises \(1-10,\) plot each complex number and find its absolute value. $$ z=4 i $$

Use the given vectors to find \(\mathbf{v} \cdot \mathbf{w}\) and \(\mathbf{v} \cdot \mathbf{v}\). $$ \mathbf{v}=3 \mathbf{i}+3 \mathbf{j}, \quad \mathbf{w}=\mathbf{i}+4 \mathbf{j} $$

In Exercises \(1-10,\) plot each complex number and find its absolute value. $$ z=3 i $$

Use the given vectors to find \(\mathbf{v} \cdot \mathbf{w}\) and \(\mathbf{v} \cdot \mathbf{v}\). $$ \mathbf{v}=5 \mathbf{i}-4 \mathbf{j}, \quad \mathbf{w}=-2 \mathbf{i}-\mathbf{j} $$

In Exercises \(1-10,\) plot each complex number and find its absolute value. $$ z=3 $$

Use the given vectors to find \(\mathbf{v} \cdot \mathbf{w}\) and \(\mathbf{v} \cdot \mathbf{v}\). $$ \mathbf{v}=7 \mathbf{i}-2 \mathbf{j}, \quad \mathbf{w}=-3 \mathbf{i}-\mathbf{j} $$

In Exercises \(1-10,\) plot each complex number and find its absolute value. $$ z=4 $$

Use the given vectors to find \(\mathbf{v} \cdot \mathbf{w}\) and \(\mathbf{v} \cdot \mathbf{v}\). $$ \mathbf{v}=-6 \mathbf{i}-5 \mathbf{j}, \quad \mathbf{w}=-10 \mathbf{i}-8 \mathbf{j} $$

In Exercises 5–12, sketch each vector as a position vector and find its magnitude. $$\mathbf{v}=3 \mathbf{i}+\mathbf{j}$$

In Exercises \(1-10,\) plot each complex number and find its absolute value. $$ z=3+2 i $$

Use the given vectors to find \(\mathbf{v} \cdot \mathbf{w}\) and \(\mathbf{v} \cdot \mathbf{v}\). $$ \mathbf{v}=-8 \mathbf{i}-3 \mathbf{j}, \quad \mathbf{w}=-10 \mathbf{i}-5 \mathbf{j} $$

In Exercises 5–12, sketch each vector as a position vector and find its magnitude. $$\mathbf{v}=2 \mathbf{i}+3 \mathbf{j}$$

In Exercises \(1-10,\) plot each complex number and find its absolute value. $$ z=2+5 i $$

Test for symmetry with respect to a. the polar axis. b. the line \(\theta=\frac{\pi}{2}\) c. the pole.$$ r=\sin \theta$$

Use the given vectors to find \(\mathbf{v} \cdot \mathbf{w}\) and \(\mathbf{v} \cdot \mathbf{v}\). $$ \mathbf{v}=5 \mathbf{i}, \quad \mathbf{w}=\mathbf{j} $$

In Exercises 5–12, sketch each vector as a position vector and find its magnitude. $$\mathbf{v}=\mathbf{i}-\mathbf{j}$$

In Exercises \(1-10,\) plot each complex number and find its absolute value. $$ z=3-i $$

Test for symmetry with respect to a. the polar axis. b. the line \(\theta=\frac{\pi}{2}\) c. the pole. $$ r=\cos \theta $$

In Exercises 5–12, sketch each vector as a position vector and find its magnitude. $$\mathbf{v}=-\mathbf{i}-\mathbf{j}$$

In Exercises \(1-10,\) plot each complex number and find its absolute value. $$ z=4-i $$

Test for symmetry with respect to a. the polar axis. b. the line \(\theta=\frac{\pi}{2}\) c. the pole. $$ r=4+3 \cos \theta $$

Let $$\mathbf{u}=2 \mathbf{i}-\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}+\mathbf{j}, \quad \text { and } \quad \mathbf{w}=\mathbf{i}+4 \mathbf{j}$$ Find each specified scalar. $$ \mathbf{u} \cdot(\mathbf{v}+\mathbf{w}) $$

Solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$a=5, b=7, C=42^{\circ}$$

In Exercises \(1-10,\) plot each complex number and find its absolute value. $$ z=-3+4 i $$

In Exercises 9–16, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$ A=44^{\circ}, B=25^{\circ}, a=12 $$

Test for symmetry with respect to a. the polar axis. b. the line \(\theta=\frac{\pi}{2}\) c. the pole. $$ r=2 \cos 2 \theta $$

Let $$\mathbf{u}=2 \mathbf{i}-\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}+\mathbf{j}, \quad \text { and } \quad \mathbf{w}=\mathbf{i}+4 \mathbf{j}$$ Find each specified scalar. $$ \mathbf{v} \cdot(\mathbf{u}+\mathbf{w}) $$

Solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$a=10, b=3, C=15^{\circ}$$

In Exercises \(1-10,\) plot each complex number and find its absolute value. $$ z=-3-4 i $$

In Exercises 9–16, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$ A=56^{\circ}, C=24^{\circ}, a=22 $$

Test for symmetry with respect to a. the polar axis. b. the line \(\theta=\frac{\pi}{2}\) c. the pole. $$ r^{2}=16 \cos 2 \theta $$

Let $$\mathbf{u}=2 \mathbf{i}-\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}+\mathbf{j}, \quad \text { and } \quad \mathbf{w}=\mathbf{i}+4 \mathbf{j}$$ Find each specified scalar. $$ \mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w} $$

Solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$b=5, c=3, A=102^{\circ}$$

In Exercises 5–12, sketch each vector as a position vector and find its magnitude. $$\mathbf{v}=-4 \mathbf{i}$$

In Exercises \(11-26,\) plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. $$ 2+2 i $$

Use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. $$ \left(2,45^{\circ}\right) $$

In Exercises 9–16, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$ B=85^{\circ}, C=15^{\circ}, b=40 $$

Test for symmetry with respect to a. the polar axis. b. the line \(\theta=\frac{\pi}{2}\) c. the pole. $$ r^{2}=16 \sin 2 \theta $$

Let $$\mathbf{u}=2 \mathbf{i}-\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}+\mathbf{j}, \quad \text { and } \quad \mathbf{w}=\mathbf{i}+4 \mathbf{j}$$ Find each specified scalar. $$ \mathbf{v} \cdot \mathbf{u}+\mathbf{v} \cdot \mathbf{w} $$

Solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$b=4, c=1, A=100^{\circ}$$

In Exercises 5–12, sketch each vector as a position vector and find its magnitude. $$v=-5 j$$

In Exercises \(11-26,\) plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. $$ 1+i \sqrt{3} $$

In Exercises 9–16, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$ A=85^{\circ}, B=35^{\circ}, c=30 $$

Use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. $$ \left(1,45^{\circ}\right) $$

Test for symmetry and then graph each polar equation. $$r=2 \cos \theta$$

Let $$\mathbf{u}=2 \mathbf{i}-\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}+\mathbf{j}, \quad \text { and } \quad \mathbf{w}=\mathbf{i}+4 \mathbf{j}$$ Find each specified scalar. $$ (4 \mathbf{u}) \cdot \mathbf{v} $$

Solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$a=6, c=5, B=50^{\circ}$$

In Exercises \(13-20\), let v be the vector from initial point \(P_{1}\) to terminal point \(P_{2} .\) Write \(\mathbf{v}\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\) $$P_{1}=(-4,-4), P_{2}=(6,2)$$

In Exercises \(11-26,\) plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. $$ -1-i $$