Chapter 9

Find the component form of the vector \(v\) whose magnitude and direction angle \(\theta\) are given. $$\|\mathbf{v}\|=5, \theta=30^{\circ}$$

Find the nth roots in polar form. $$1-\sqrt{3} i ; \quad n=3$$

In Exercises \(25-36,\) express the number in the form \(a+b i\). $$4\left(\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}\right)$$

Find proju \(v\) and proju u. $$\mathbf{u}=3 \mathbf{i}-5 \mathbf{j}, \mathbf{v}=6 \mathbf{i}+2 \mathbf{j}$$

Find the component form of the vector \(v\) whose magnitude and direction angle \(\theta\) are given. $$\|\mathbf{v}\|=10, \theta=225^{\circ}$$

Find the nth roots in polar form. $$8 \sqrt{3}+8 i ; \quad n=4$$

In Exercises \(25-36,\) express the number in the form \(a+b i\). $$5\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)$$

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

Find the component form of the vector \(v\) whose magnitude and direction angle \(\theta\) are given. $$\|\mathbf{v}\|=20, \theta=120^{\circ}$$

Find the nth roots in polar form. $$-16 \sqrt{2}-16 \sqrt{2} i ; \quad n=5$$

In Exercises \(25-36,\) express the number in the form \(a+b i\). $$2\left(\cos \frac{7 \pi}{6}+i \sin \frac{7 \pi}{6}\right)$$

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

Find the component form of the vector \(v\) whose magnitude and direction angle \(\theta\) are given. $$\|\mathbf{v}\|=6, \theta=40^{\circ}$$

Solve the given equation in the complex number system. $$x^{6}=-1$$

In Exercises \(25-36,\) express the number in the form \(a+b i\). $$1.5\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right)$$

Find proju \(v\) and proju u. $$\mathbf{u}=5 \mathbf{i}+\mathbf{j}, \mathbf{v}=-2 \mathbf{i}+3 \mathbf{j}$$

Solve the given equation in the complex number system. $$x^{6}+64=0$$

In Exercises \(25-36,\) express the number in the form \(a+b i\). $$5(\cos 3+i \sin 3)$$

find comp, \(u\) $$\mathbf{u}=10 \mathbf{i}+4 \mathbf{j}, \mathbf{v}=3 \mathbf{i}-2 \mathbf{j}$$

Find the component form of the vector \(v\) whose magnitude and direction angle \(\theta\) are given. $$\|\mathbf{v}\|=1 / 2, \boldsymbol{\theta}=250^{\circ}$$

Solve the given equation in the complex number system. $$x^{3}=i$$

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

find comp, \(u\) $$\mathbf{u}=\mathbf{i}-2 \mathbf{j}, \mathbf{v}=3 \mathbf{i}+\mathbf{j}$$

Find the component form of the vector \(v\) whose magnitude and direction angle \(\theta\) are given. $$\|\mathbf{v}\|=3, \boldsymbol{\theta}=310^{\circ}$$

Solve the given equation in the complex number system. $$x^{4}=i$$

In Exercises \(25-36,\) express the number in the form \(a+b i\). $$3(\cos 5+i \sin 5)$$

find comp, \(u\) $$\mathbf{u}=3 \mathbf{i}+2 \mathbf{j}, \mathbf{v}=-\mathbf{i}+3 \mathbf{j}$$

Find the magnitude and direction angle of the vector \(\boldsymbol{v}\). $$\mathbf{v}=\langle 4,4\rangle$$

Solve the given equation in the complex number system. $$x^{3}+27 i=0$$

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

find comp, \(u\) $$\mathbf{u}=\mathbf{i}+\mathbf{j}, \mathbf{v}=-3 \mathbf{i}-2 \mathbf{j}$$

Find the magnitude and direction angle of the vector \(\boldsymbol{v}\). $$\mathbf{v}=\langle 5,5 \sqrt{3}\rangle$$

Solve the given equation in the complex number system. $$x^{6}+729=0$$

In Exercises \(25-36,\) express the number in the form \(a+b i\). $$2(\cos 1.5+i \sin 1.5)$$

Let \(\boldsymbol{u}=\langle a, b\rangle, \boldsymbol{v}=\langle c, d\rangle,\) and \(\boldsymbol{w}=\langle r, s\rangle\) Verify that the given property of dot products is valid by calculating the quantities on each side of the equal sign. $$\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w}$$

Find the magnitude and direction angle of the vector \(\boldsymbol{v}\). $$\mathbf{v}=\langle-8,0\rangle$$

Solve the given equation in the complex number system. $$x^{5}-243 i=0$$

In Exercises \(37-52,\) express the number in polar form. $$3+3 i$$

Let \(\boldsymbol{u}=\langle a, b\rangle, \boldsymbol{v}=\langle c, d\rangle,\) and \(\boldsymbol{w}=\langle r, s\rangle\) Verify that the given property of dot products is valid by calculating the quantities on each side of the equal sign. $$k \mathbf{u} \cdot \mathbf{v}=k(\mathbf{u} \cdot \mathbf{v})=\mathbf{u} \cdot k \mathbf{v}$$

Find the magnitude and direction angle of the vector \(\boldsymbol{v}\). $$\mathbf{v}=\langle 4,5\rangle$$

In Exercises \(37-52,\) express the number in polar form. $$5-5 i$$

Let \(\boldsymbol{u}=\langle a, b\rangle, \boldsymbol{v}=\langle c, d\rangle,\) and \(\boldsymbol{w}=\langle r, s\rangle\) Verify that the given property of dot products is valid by calculating the quantities on each side of the equal sign. $$0 \cdot \mathbf{u}=0$$

Find the magnitude and direction angle of the vector \(\boldsymbol{v}\). $$\mathbf{v}=6 \mathbf{j}$$

Solve the given equation in the complex number system. $$x^{4}=-1+\sqrt{3} i$$

In Exercises \(37-52,\) express the number in polar form. $$2+2 \sqrt{3} i$$

Suppose \(\mathbf{u}=\langle a, b\rangle\) and \(\mathbf{v}=\langle c, d\rangle\) are nonzero parallel vectors. (a) If \(c \neq 0,\) show that \(\mathbf{u}\) and \(\mathbf{v}\) lie on the same nonvertical straight line through the origin. (b) If \(c \neq 0,\) show that \(\mathbf{v}=\frac{a}{c} \mathbf{u}\) (that is, \(\mathbf{v}\) is a scalar multiple of \(\mathbf{u}\) ). [Hint: The equation of the line on which \(\mathbf{u}\) and v lie is \(y=m x\) for some constant \(m\) (why?), which implies that \(b=m a\) and \(d=m c .\) ] (c) If \(c=0,\) show that \(\mathbf{v}\) is a scalar multiple of \(\mathbf{u} .[\) Hint: If \(c=0,\) then \(a=0\) (why?), and hence, \(b \neq 0\) (otherwise, \(\mathbf{u}=\mathbf{0}) .]\)

Find the magnitude and direction angle of the vector \(\boldsymbol{v}\). $$\mathbf{v}=4 \mathbf{i}-8 \mathbf{j}$$

Solve the given equation in the complex number system. $$x^{4}=-8-8 \sqrt{3} i$$

In Exercises \(37-52,\) express the number in polar form. $$5 \sqrt{3}+5 i$$

Prove the Angle Theorem in the case when \(\theta\) is 0 or \(\pi\)