Chapter 9

1-8 Find \((a) \mathbf{u} \cdot \mathbf{v}\) and \((\mathbf{b})\) the angle between \(\mathbf{u}\) and \(\mathbf{v}\) to the nearest degree. $$\mathbf{u}=\langle 2,0\rangle, \quad \mathbf{v}=\langle 1,1\rangle$$

\(\mathbf{1}-8\) Graph the complex number and find its modulus. $$ 4 i $$

Plot the point that has the given polar coordinates. $$ (4, \pi / 4) $$

1-8 Find \((a) \mathbf{u} \cdot \mathbf{v}\) and \((\mathbf{b})\) the angle between \(\mathbf{u}\) and \(\mathbf{v}\) to the nearest degree. $$\mathbf{u}=\mathbf{i}+\sqrt{3} \mathbf{j}, \quad \mathbf{v}=-\sqrt{3} \mathbf{i}+\mathbf{j}$$

\(\mathbf{1}-8\) Graph the complex number and find its modulus. $$ -3 i $$

Plot the point that has the given polar coordinates. $$ (1,0) $$

1-8 Find \((a) \mathbf{u} \cdot \mathbf{v}\) and \((\mathbf{b})\) the angle between \(\mathbf{u}\) and \(\mathbf{v}\) to the nearest degree. $$\mathbf{u}=\langle 2,7\rangle, \quad \mathbf{v}=\langle 3,1\rangle$$

\(\mathbf{1}-8\) Graph the complex number and find its modulus. $$ -2 $$

Plot the point that has the given polar coordinates. $$ (6,-7 \pi / 6) $$

1-8 Find \((a) \mathbf{u} \cdot \mathbf{v}\) and \((\mathbf{b})\) the angle between \(\mathbf{u}\) and \(\mathbf{v}\) to the nearest degree. $$\mathbf{u}=\langle- 6,6\rangle, \quad \mathbf{v}=\langle 1,-1\rangle$$

\(\mathbf{1}-8\) Graph the complex number and find its modulus. $$ 6 $$

Plot the point that has the given polar coordinates. $$ (3,-2 \pi / 3) $$

1-8 Find \((a) \mathbf{u} \cdot \mathbf{v}\) and \((\mathbf{b})\) the angle between \(\mathbf{u}\) and \(\mathbf{v}\) to the nearest degree. $$\mathbf{u}=\langle 3,-2\rangle, \quad \mathbf{v}=\langle 1,2\rangle$$

\(\mathbf{1}-8\) Graph the complex number and find its modulus. $$ 5+2 i $$

Plot the point that has the given polar coordinates. $$ (-2,4 \pi / 3) $$

1-8 Find \((a) \mathbf{u} \cdot \mathbf{v}\) and \((\mathbf{b})\) the angle between \(\mathbf{u}\) and \(\mathbf{v}\) to the nearest degree. $$\mathbf{u}=2 \mathbf{i}+\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}-2 \mathbf{j}$$

\(1-8\) Graph the complex number and find its modulus. $$ 7-3 i $$

Plot the point that has the given polar coordinates. $$ (-5,-17 \pi / 6) $$

1-8 Find \((a) \mathbf{u} \cdot \mathbf{v}\) and \((\mathbf{b})\) the angle between \(\mathbf{u}\) and \(\mathbf{v}\) to the nearest degree. $$\mathbf{u}=-5 \mathbf{j}, \quad \mathbf{v}=-\mathbf{i}-\sqrt{3} \mathbf{j}$$

\(1-8\) Graph the complex number and find its modulus. $$ \sqrt{3}+i $$

7-14 Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2 .\) $$r=2-\sin \theta$$

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\). $$ (3, \pi / 2) $$

1-8 Find \((a) \mathbf{u} \cdot \mathbf{v}\) and \((\mathbf{b})\) the angle between \(\mathbf{u}\) and \(\mathbf{v}\) to the nearest degree. $$\mathbf{u}=\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=\mathbf{i}-\mathbf{j}$$

\(1-8\) Graph the complex number and find its modulus. $$ -1-\frac{\sqrt{3}}{3} i $$

7-14 Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2 .\) $$r=4+8 \cos \theta$$

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\). $$ (2,3 \pi / 4) $$

9–14 Determine whether the given vectors are orthogonal. $$\mathbf{u}=\langle 6,4\rangle, \quad \mathbf{v}=\langle- 2,3\rangle$$

$$ \frac{3+4 i}{5} $$

7-14 Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2 .\) $$r=3 \sec \theta$$

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\). $$ (-1,7 \pi / 6) $$

9–14 Determine whether the given vectors are orthogonal. $$\mathbf{u}=\langle 0,-5\rangle, \quad \mathbf{v}=\langle 4,0\rangle$$

$$ \frac{-\sqrt{2}+i \sqrt{2}}{2} $$

7-14 Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2 .\) $$r=5 \cos \theta \csc \theta$$

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\). $$ (-2,-\pi / 3) $$

Express the vector with initial point \(P\) and terminal point \(Q\) in component form. $$ P(3,2), \quad Q(8,9) $$

9–14 Determine whether the given vectors are orthogonal. $$\mathbf{u}=\langle- 2,6\rangle, \quad \mathbf{v}=\langle 4,2\rangle$$

\(11-12\) a sketch the complex number \(z,\) and also sketch \(2 z,-z\) and \(\frac{1}{2} z\) on the same complex plane. $$ z=1+i $$

7-14 Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2 .\) $$r=\frac{4}{3-2 \sin \theta}$$

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\). $$ (-5,0) $$

Express the vector with initial point \(P\) and terminal point \(Q\) in component form. $$ P(1,1), \quad Q(9,9) $$

9–14 Determine whether the given vectors are orthogonal. $$\mathbf{u}=2 \mathbf{i}, \quad \mathbf{v}=-7 \mathbf{j}$$

\(11-12\) a sketch the complex number \(z,\) and also sketch \(2 z,-z\) and \(\frac{1}{2} z\) on the same complex plane. $$ z=-1+i \sqrt{3} $$

7-14 Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2 .\) $$r=\frac{5}{1+3 \cos \theta}$$

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\). $$ (3,1) $$

Express the vector with initial point \(P\) and terminal point \(Q\) in component form. $$ P(5,3), \quad Q(1,0) $$

9–14 Determine whether the given vectors are orthogonal. $$\mathbf{u}=2 \mathbf{i}-8 \mathbf{j}, \quad \mathbf{v}=-12 \mathbf{i}-3 \mathbf{j}$$

\(13-14\) : Sketch the complex number \(z\) and its complex conjugate \(z\) on the same complex plane. $$ z=8+2 i $$

7-14 Test the polar equation for symmetry with respect to the polar axis, the pole, and the line \(\theta=\pi / 2 .\) $$r^{2}=4 \cos 2 \theta$$

Express the vector with initial point \(P\) and terminal point \(Q\) in component form. $$ P(-1,3), \quad Q(-6,-1) $$

9–14 Determine whether the given vectors are orthogonal. $$\mathbf{u}=4 \mathbf{i}, \quad \mathbf{v}=-\mathbf{i}+3 \mathbf{j}$$