Chapter 9

Find the magnitude and direction angle of the vector \(\boldsymbol{v}\). $$\mathbf{v}=-2 \mathbf{i}+8 \mathbf{j}$$

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

If \(\mathbf{u}\) and \(\mathbf{v}\) are nonzero vectors such that \(\mathbf{u} \cdot \mathbf{v}=0,\) show that u and \(v\) are orthogonal. \([\text {Hint} \text { : If } \theta \text { is the angle between } \mathbf{u}\) and \(\mathbf{v}, \text { what is } \cos \theta \text { and what does this say about } \theta ?]\)

Find the magnitude and direction angle of the vector \(\boldsymbol{v}\). $$\mathbf{v}=-15 \mathbf{i}-10 \mathbf{j}$$

Represent the roots of unity graphically. Then use the trace feature to obtain approximations of the form \(a+\) bi for each root (round to four places). Fifth roots of unity

In Exercises \(37-52,\) express the number in polar form. $$-4-4 \sqrt{3} i$$

Show that (1,2),(3,4),(5,2) are the vertices of a right triangle by considering the sides of the triangle as vectors.

Find a unit vector that has the same direction as \(v\). $$\langle 4,-5\rangle$$

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

Find a number \(x\) such that the angle between the vectors \langle 1,1\rangle and \(\langle x, 1\rangle\) is \(\pi / 4\) radians.

Find a unit vector that has the same direction as \(v\). $$-7 \mathbf{i}+8 \mathbf{j}$$

In Exercises \(37-52,\) express the number in polar form. $$2 \sqrt{5}-2 \sqrt{5} i$$

Find nonzero vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) such that \(\mathbf{u} \cdot \mathbf{v}=\mathbf{u} \cdot \mathbf{w}\) and \(\mathbf{v} \neq \mathbf{w}\) and neither \(\mathbf{v}\) nor \(\mathbf{w}\) is orthogonal to \(\mathbf{u}\)

Find a unit vector that has the same direction as \(v\). $$5 \mathbf{i}+10 \mathbf{j}$$

In Exercises \(37-52,\) express the number in polar form. $$3+4 i$$

If \(\mathbf{u}\) and \(\mathbf{v}\) are nonzero vectors, show that the vectors \(\|\mathbf{u}\| \mathbf{v}+\|\mathbf{v}\| \mathbf{u}\) and \(\|\mathbf{u}\| \mathbf{v}-\|\mathbf{v}\| \mathbf{u}\) are orthogonal.

Find a unit vector that has the same direction as \(v\). $$-3 \mathbf{i}-9 \mathbf{j}$$

In Exercises \(37-52,\) express the number in polar form. $$-4+3 i$$

A 600 -pound trailer is on an inclined ramp that makes a \(30^{\circ}\) angle with the horizontal. Find the force required to keep it from rolling down the ramp, assuming that the only force that must be overcome is that due to gravity.

An object at the origin is acted upon by two forces, \(u\) and \(v,\) with direction angle \(\theta_{u}\) and \(\theta_{w}\) respectively. Find the direction and magnitude of the resultant force. $$\mathbf{u}=30 \text { pounds, } \theta_{u}=0^{\circ} ; \mathbf{v}=90 \text { pounds, } \theta_{v}=60^{\circ}$$

Solve the equation \(x^{3}+x^{2}+x+1=0 .\) [ Hint: First find the quotient when \(x^{4}-1\) is divided by \(x-1\) and then consider solutions of \(\left.x^{4}-1=0 .\right]\)

In Exercises \(37-52,\) express the number in polar form. $$5-12 i$$

An object at the origin is acted upon by two forces, \(u\) and \(v,\) with direction angle \(\theta_{u}\) and \(\theta_{w}\) respectively. Find the direction and magnitude of the resultant force. $$\mathbf{u}=6 \text { pounds }, \theta_{\mathbf{u}}=45^{\circ} ; \mathbf{v}=6 \text { pounds }, \theta_{\mathbf{v}}=120^{\circ}$$

Solve the equation \(x^{4}+x^{3}+x^{2}+x+1=0 .\) [Hint: Consider \(\left.x^{5}-1 \text { and } x-1 \text { and see Exercise } 47 .\right]\)

Find the work done by a constant force \(\boldsymbol{F}\) as the point of application of \(\boldsymbol{F}\) moves along the vector \(\overrightarrow{P Q}\). $$\mathbf{F}=2 \mathbf{i}+5 \mathbf{j}, P=(0,0), Q=(4,1)$$

An object at the origin is acted upon by two forces, \(u\) and \(v,\) with direction angle \(\theta_{u}\) and \(\theta_{w}\) respectively. Find the direction and magnitude of the resultant force. $$\mathbf{u}=12 \text { newtons, } \theta_{\mathbf{u}}=130^{\circ} ; \mathbf{v}=20 \text { newtons } \theta_{\mathbf{v}}=250^{\circ}$$

Solve \(x^{5}+x^{4}+x^{3}+x^{2}+x+1=0 .\) IHint: Consider \(\left.x^{6}-1 \text { and } x-1 \text { and see Exercise } 47 .\right]\)

In Exercises \(37-52,\) express the number in polar form. $$1+2 i$$

Find the work done by a constant force \(\boldsymbol{F}\) as the point of application of \(\boldsymbol{F}\) moves along the vector \(\overrightarrow{P Q}\). $$\mathbf{F}=\mathbf{i}-2 \mathbf{j}, P=(0,0), Q=(-5,2)$$

An object at the origin is acted upon by two forces, \(u\) and \(v,\) with direction angle \(\theta_{u}\) and \(\theta_{w}\) respectively. Find the direction and magnitude of the resultant force. $$\mathbf{u}=30 \text { newtons, } \theta_{\mathbf{u}}=300^{\circ} ; \mathbf{v}=80 \text { newtons, } \theta_{\mathbf{v}}=40^{\circ}$$

What do you think are the solutions of \(x^{n-1}+x^{n-2}+\cdots+x^{3}+x^{2}+x+1=0 ?\) (Sce Exer- cises \(47-49 .)\)

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

Find the work done by a constant force \(\boldsymbol{F}\) as the point of application of \(\boldsymbol{F}\) moves along the vector \(\overrightarrow{P Q}\). \(\mathbf{F}=2 \mathbf{i}+3 \mathbf{j}, P=(2,3), Q=(5,9)[\) Hint: Find the compo- nent form of \(\overrightarrow{P Q}\).]

If forces \(\boldsymbol{u}_{1}, \boldsymbol{u}_{2}, \ldots, \boldsymbol{u}_{k}\) act on an object at the origin, the resultant force is the sum \(u_{1}+u_{2}+\cdots+u_{k} .\) The forces are said to be in equilibrium if their resultant force is \(0 .\) In Exercises 51 and \(52,\) find the resultant force and find an additional force \(v\) that, if added to the system, produces equilibrium. $$\mathbf{u}_{1}=\langle 2,5\rangle, \mathbf{u}_{2}=\langle-6,1\rangle, \mathbf{u}_{3}=\langle-4,-8\rangle$$

In Exercises \(37-52,\) express the number in polar form. $$-\frac{5}{2}+\frac{7}{2} i$$

Find the work done by a constant force \(\boldsymbol{F}\) as the point of application of \(\boldsymbol{F}\) moves along the vector \(\overrightarrow{P Q}\). $$\mathbf{F}=5 \mathbf{i}+\mathbf{j}, P=(-1,2), Q=(4,-3)$$

If forces \(\boldsymbol{u}_{1}, \boldsymbol{u}_{2}, \ldots, \boldsymbol{u}_{k}\) act on an object at the origin, the resultant force is the sum \(u_{1}+u_{2}+\cdots+u_{k} .\) The forces are said to be in equilibrium if their resultant force is \(0 .\) In Exercises 51 and \(52,\) find the resultant force and find an additional force \(v\) that, if added to the system, produces equilibrium. $$\mathbf{u}_{1}=\langle 3,7\rangle, \mathbf{u}_{2}=\langle 8,-2\rangle, \mathbf{u}_{3}=\langle-9,0\rangle, \mathbf{u}_{4}=\langle-5,4\rangle$$

In Exercises \(37-52,\) express the number in polar form. $$\sqrt{5}+\sqrt{11} i$$

Let \(\boldsymbol{u}=\langle a, b\rangle\) and \(\boldsymbol{v}=\langle c, d\rangle,\) and let \(r\) and s be scalars. Prove that the stated property holds by calculating the vector on each side of the equal sign. $$\mathbf{v}+\mathbf{0}=\mathbf{v}=\mathbf{0}+\mathbf{v}$$

In Exercises \(53-64,\) perform the indicated multiplication or division. Express your answer in both polar form \(r(\cos \theta+i \sin \theta)\) and rectangular form \(a+b i\). $$\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right) \cdot(\cos \pi+i \sin \pi)$$

A child pulls a wagon along a level sidewalk by exerting a force of 18 pounds on the wagon handle, which makes an angle of \(25^{\circ}\) with the horizontal. How much work is done in pulling the wagon 200 feet?

Let \(\boldsymbol{u}=\langle a, b\rangle\) and \(\boldsymbol{v}=\langle c, d\rangle,\) and let \(r\) and s be scalars. Prove that the stated property holds by calculating the vector on each side of the equal sign. $$\mathbf{v}+(-\mathbf{v})=\mathbf{0}$$

In Exercises \(53-64,\) perform the indicated multiplication or division. Express your answer in both polar form \(r(\cos \theta+i \sin \theta)\) and rectangular form \(a+b i\). $$2\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right) \cdot 5\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)$$

Let \(\boldsymbol{u}=\langle a, b\rangle\) and \(\boldsymbol{v}=\langle c, d\rangle,\) and let \(r\) and s be scalars. Prove that the stated property holds by calculating the vector on each side of the equal sign. $$r(\mathbf{u}+\mathbf{v})=r \mathbf{u}+r \mathbf{v}$$

In Exercises \(53-64,\) perform the indicated multiplication or division. Express your answer in both polar form \(r(\cos \theta+i \sin \theta)\) and rectangular form \(a+b i\). $$4\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right) \cdot 3\left(\cos \frac{\pi}{12}+i \sin \frac{\pi}{12}\right)$$

Let \(\boldsymbol{u}=\langle a, b\rangle\) and \(\boldsymbol{v}=\langle c, d\rangle,\) and let \(r\) and s be scalars. Prove that the stated property holds by calculating the vector on each side of the equal sign. $$(r+s) \mathbf{v}=r \mathbf{v}+s \mathbf{v}$$

In Exercises \(53-64,\) perform the indicated multiplication or division. Express your answer in both polar form \(r(\cos \theta+i \sin \theta)\) and rectangular form \(a+b i\). $$\left(\cos \frac{\pi}{12}+i \sin \frac{\pi}{12}\right) \cdot 2\left(\cos \frac{7 \pi}{12}+i \sin \frac{7 \pi}{12}\right)$$

Let \(\boldsymbol{u}=\langle a, b\rangle\) and \(\boldsymbol{v}=\langle c, d\rangle,\) and let \(r\) and s be scalars. Prove that the stated property holds by calculating the vector on each side of the equal sign. $$(r s) \mathbf{v}=r(s \mathbf{v})=s(r \mathbf{v})$$

In Exercises \(53-64,\) perform the indicated multiplication or division. Express your answer in both polar form \(r(\cos \theta+i \sin \theta)\) and rectangular form \(a+b i\). $$3\left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right) \cdot 12\left(\cos \frac{3 \pi}{8}+i \sin \frac{3 \pi}{8}\right)$$

In Exercises \(53-64,\) perform the indicated multiplication or division. Express your answer in both polar form \(r(\cos \theta+i \sin \theta)\) and rectangular form \(a+b i\). $$12\left(\cos \frac{11 \pi}{12}+i \sin \frac{11 \pi}{12}\right) \cdot \frac{7}{2}\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)$$