Chapter 9

Calculate the product by expressing the number in polar form and using DeMoivre's Theorem. Express your answer in the form \(a+b i\). $$(-1+\sqrt{3} i)^{8}$$

In Exercises \(9-14,\) find the absolute value. $$\left|i^{7}\right|$$

Find \(u+v, v-u,\) and \(2 u-3 v\). $$\mathbf{u}=2\langle-2,5\rangle, \mathbf{v}=\frac{1}{4}\langle-7,12\rangle$$

Find the indicated roots of unity and express your answers in the form \(a+b i\). Fourth roots of unity

Give an example of complex numbers \(z\) and \(w\) such that \(|z+w| \neq|z|+|w|\)

Find the angle between the two vectors. $$2 \mathbf{j}, 4 \mathbf{i}+\mathbf{j}$$

Find \(u+v, v-u,\) and \(2 u-3 v\). $$\mathbf{u}=\mathbf{i}-\mathbf{j}, \mathbf{v}=2 \mathbf{i}+\mathbf{j}$$

Find the indicated roots of unity and express your answers in the form \(a+b i\). Sixth roots of unity

If \(z=3-4 i,\) find \(|z|^{2}\) and \(z \bar{z},\) where \(\bar{z}\) is the conjugate of \(z\) (see page 323 ).

Find the angle between the two vectors. $$\sqrt{2} \mathbf{i}+\sqrt{2} \mathbf{j}, \mathbf{i}-\mathbf{j}$$

Find \(u+v, v-u,\) and \(2 u-3 v\). $$\mathbf{u}=8 \mathbf{i}, \mathbf{v}=2(3 \mathbf{i}-2 \mathbf{j})$$

Find the nth roots in polar form. $$36\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right) ; \quad n=2$$

In Exercises \(17-24,\) sketch the graph of the equation in the complex plane (z denotes a complex number of the form a \(+b i\) ). \(|z|=4[\text {Hint}:\) The graph consists of all points that lie 4 units from the origin. \(]\)

Find the angle between the two vectors. $$3 \mathbf{i}-5 \mathbf{j},-2 \mathbf{i}+3 \mathbf{j}$$

Find \(u+v, v-u,\) and \(2 u-3 v\). $$\mathbf{u}=-4(-\mathbf{i}+\mathbf{j}), \mathbf{v}=-3 \mathbf{i}$$

Find the nth roots in polar form. $$64\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right) ; \quad n=2$$

In Exercises \(17-24,\) sketch the graph of the equation in the complex plane (z denotes a complex number of the form a \(+b i\) ). $$|z|=1$$

Determine whether the given vectors are parallel, orthogonal, or neither. $$\langle 2,6\rangle,\langle 3,-1\rangle$$

Find \(u+v, v-u,\) and \(2 u-3 v\). $$\mathbf{u}=-\left(2 \mathbf{i}+\frac{3}{2} \mathbf{j}\right), \mathbf{v}=\frac{3}{4} \mathbf{i}$$

Find the nth roots in polar form. $$64\left(\cos \frac{\pi}{5}+i \sin \frac{\pi}{5}\right) ; \quad n=3$$

In Exercises \(17-24,\) sketch the graph of the equation in the complex plane (z denotes a complex number of the form a \(+b i\) ). \(|z-1|=10[\text {Hint: } 1 \text { corresponds to }(1,0)\) in the complex plane. What does the equation say about the distance from \(z\) to \(1 ?]\)

Determine whether the given vectors are parallel, orthogonal, or neither. $$\langle-5,3\rangle,\langle 2,6\rangle$$

Find \(u+v, v-u,\) and \(2 u-3 v\). $$\mathbf{u}=\sqrt{2} \mathbf{j}, \mathbf{v}=\sqrt{3} \mathbf{i}$$

Find the nth roots in polar form. $$8\left(\cos \frac{\pi}{10}+i \sin \frac{\pi}{10}\right) ; \quad n=3$$

In Exercises \(17-24,\) sketch the graph of the equation in the complex plane (z denotes a complex number of the form a \(+b i\) ). $$|z+3|=1$$

Determine whether the given vectors are parallel, orthogonal, or neither. $$\langle 9,-6\rangle,\langle-6,4\rangle$$

Find the nth roots in polar form. $$81\left(\cos \frac{\pi}{12}+i \sin \frac{\pi}{12}\right) ; \quad n=4$$

In Exercises \(17-24,\) sketch the graph of the equation in the complex plane (z denotes a complex number of the form a \(+b i\) ). $$|z-2 i|=4$$

Determine whether the given vectors are parallel, orthogonal, or neither. $$-\mathbf{i}+2 \mathbf{j}, 2 \mathbf{i}-4 \mathbf{j}$$

Find the nth roots in polar form. $$16\left(\cos \frac{\pi}{7}+i \sin \frac{\pi}{7}\right) ; \quad n=5$$

In Exercises \(17-24,\) sketch the graph of the equation in the complex plane (z denotes a complex number of the form a \(+b i\) ). $$|z-3 i+2|=9[\text {Hint: Rewrite it as }|z-(-2+3 i)|=9 .]$$

Determine whether the given vectors are parallel, orthogonal, or neither. $$2 \mathbf{i}-2 \mathbf{j}, 5 \mathbf{i}+8 \mathbf{j}$$

Find the nth roots in polar form. $$-1 ; \quad n=5$$

In Exercises \(17-24,\) sketch the graph of the equation in the complex plane (z denotes a complex number of the form a \(+b i\) ). \(\operatorname{Re}(z)=2\) [The real part of the complex number \(z=a+b i\) is defined to be the number \(a\) and is denoted \(\operatorname{Re}(z) .]\)

Determine whether the given vectors are parallel, orthogonal, or neither. $$6 i-4 j, 2 i+3 j$$

Find the components of the given vector, where \(\boldsymbol{u}=\boldsymbol{i}-2 \boldsymbol{j}, \boldsymbol{v}=3 \boldsymbol{i}+\boldsymbol{j}, \boldsymbol{w}=-4 \boldsymbol{i}+\boldsymbol{j}\) $$-2 \mathbf{u}+3 \mathbf{v}$$

Find the nth roots in polar form. $$1 ; \quad n=7$$

In Exercises \(17-24,\) sketch the graph of the equation in the complex plane (z denotes a complex number of the form a \(+b i\) ). \(\operatorname{Im}(z)=-5 / 2\) [The imaginary part of \(z=a+b i\) is defined to be the number \(b \text { ( not bi) and is denoted } \operatorname{Im}(z) .]\)

Find a real number \(k\) such that the two vectors are orthogonal. $$2 \mathbf{i}+3 \mathbf{j}, 3 \mathbf{i}-k \mathbf{j}$$

Find the components of the given vector, where \(\boldsymbol{u}=\boldsymbol{i}-2 \boldsymbol{j}, \boldsymbol{v}=3 \boldsymbol{i}+\boldsymbol{j}, \boldsymbol{w}=-4 \boldsymbol{i}+\boldsymbol{j}\) $$\frac{1}{4}(8 u+4 v-w)$$

Find the nth roots in polar form. $$i ; \quad n=5$$

In Exercises \(25-36,\) express the number in the form \(a+b i\). $$2\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)$$

Find a real number \(k\) such that the two vectors are orthogonal. $$-3 \mathbf{i}+\mathbf{j}, 2 k \mathbf{i}-4 \mathbf{j}$$

Find the components of the given vector, where \(\boldsymbol{u}=\boldsymbol{i}-2 \boldsymbol{j}, \boldsymbol{v}=3 \boldsymbol{i}+\boldsymbol{j}, \boldsymbol{w}=-4 \boldsymbol{i}+\boldsymbol{j}\) $$3(\mathbf{u}-2 \mathbf{v})-6 \mathbf{w}$$

In Exercises \(25-36,\) express the number in the form \(a+b i\). $$3\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)$$

Find a real number \(k\) such that the two vectors are orthogonal. $$\mathbf{i}-\mathbf{j}, k \mathbf{i}+\sqrt{2} \mathbf{j}$$

Find the component form of the vector \(v\) whose magnitude and direction angle \(\theta\) are given. $$\|\mathbf{v}\|=4, \theta=0^{\circ}$$

Find the nth roots in polar form. $$1+i ; \quad n=2$$

In Exercises \(25-36,\) express the number in the form \(a+b i\). $$\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}$$

Find a real number \(k\) such that the two vectors are orthogonal. $$-4 \mathbf{i}+5 \mathbf{j}, 2 \mathbf{i}+2 k \mathbf{j}$$