Chapter 6

In Exercises 67-74, find the component form of \(\mathbf{v}\) given its magnitude and the angle it makes with the positive \(x\)-axis. Sketch \(\mathbf{v}\). Magnitude - ||\(\mathbf{v}\)|| \(= \frac{7}{2}\) Angle - \(\theta = 150^{\circ}\)

CIRCUMSCRIBED AND INSCRIBED CIRCLES In Exercises 68 and 69, use the results of Exercise 67. Find the length of the largest circular running track that can be built on a triangular piece of property with sides of lengths 200 feet, 250 feet, and 325 feet.

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$ 4(1-\sqrt{3} i)^{3} $$

In Exercises \(67-70,\) find two vectors in opposite directions that are orthogonal to the vector \(\mathbf{u}\) . There are many correct answers.) $$ \mathbf{u}=-\frac{5}{2} \mathbf{i}-3 \mathbf{j} $$

In Exercises 67-74, find the component form of \(\mathbf{v}\) given its magnitude and the angle it makes with the positive \(x\)-axis. Sketch \(\mathbf{v}\). Magnitude - ||\(\mathbf{v}\)|| \(= \frac{3}{4}\) Angle - \(\theta = 150^{\circ}\)

THINK ABOUT IT What familiar formula do you obtain when you use the third form of the Law of Cosines \(c^2 = a^2 + b^2 - 2ab\ \cos\ C\), and you let \(C = 90^{\circ}\)? What is the relationship between the Law of Cosines and this formula?

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$ \left[5\left(\cos 20^{\circ}+i \sin 20^{\circ}\right)\right]^{3} $$

WORK In Exercises 71 and 72, find the work done in moving a particle from \(P\) to \(Q\) if the magnitude and direction of the force are given by \(\mathbf{v}\). \(P(0, 0)\), \(Q(4, 7)\), \(\mathbf{v} = \langle 1, 4 \rangle\)

In Exercises 67-74, find the component form of \(\mathbf{v}\) given its magnitude and the angle it makes with the positive \(x\)-axis. Sketch \(\mathbf{v}\). Magnitude - ||\(\mathbf{v}\)|| \(= 2\sqrt{3}\) Angle - \(\theta = 45^{\circ}\)

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \(4(1\ -\ \sqrt{3}i)^3\)

WORK In Exercises 71 and 72, find the work done in moving a particle from \(P\) to \(Q\) if the magnitude and direction of the force are given by \(\mathbf{v}\). \(P(1, 3)\), \(Q(-3, 5)\), \(\mathbf{v} = -2\mathbf{i} +3 \mathbf{j}\)

In Exercises 67-74, find the component form of \(\mathbf{v}\) given its magnitude and the angle it makes with the positive \(x\)-axis. Sketch \(\mathbf{v}\). Magnitude - ||\(\mathbf{v}\)|| \(= 4\sqrt{3}\) Angle - \(\theta = 90^{\circ}\)

WRITING Describe how the Law of Cosines can be used to solve the ambiguous case of the oblique triangle \(ABC\), where \(a = 12\) feet, \(b = 30\) feet, and \(A = 20^{\circ}\). Is the result the same as when the Law of Sines is used to solve the triangle? Describe the advantages and the disadvantages of each method.

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \([5(\cos\ 20^{\circ} + i\ \sin\ 20^{\circ})]^3\)

REVENUE The vector \(\mathbf{u} = \langle 4600, 5260 \rangle\) gives the numbers of units of two models of cellular phones produced by a telecommunications company. The vector \(\mathbf{v} = \langle 79.99, 99.99 \rangle\) gives the prices (in dollars) of the two models of cellular phones, respectively. (a) Find the dot product \(\mathbf{u} \cdot \mathbf{v}\) and interpret the result in the context of the problem. (b) Identify the vector operation used to increase the prices by 5%.

In Exercises 67-74, find the component form of \(\mathbf{v}\) given its magnitude and the angle it makes with the positive \(x\)-axis. Sketch \(\mathbf{v}\). Magnitude - ||\(\mathbf{v}\)|| \(= 3\) Angle - \(\mathbf{v}\) in the direction \(3\mathbf{i} + 4\mathbf{j}\)

WRITING In Exercise 72, the Law of Cosines was used to solve a triangle in the two-solution case of SSA.Can the Law of Cosines be used to solve the no- solution and single-solution cases of SSA? Explain.

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \([3(\cos\ 60^{\circ} + i\ \sin\ 60^{\circ})]^4\)

REVENUE The vector \(\mathbf{u} = \langle 3140, 2750 \rangle\) gives the numbers of hamburgers and hot dogs, respectively, sold at a fast-food stand in one month. The vector \(\mathbf{v} = \langle 2.25, 1.75 \rangle\) gives the prices (in dollars) of the food items. (a) Find the dot product \(\mathbf{u} \cdot \mathbf{v}\) and interpret the result in the context of the problem. (b) Identify the vector operation used to increase the prices by 2.5%.

CAPSTONE Determine whether the Law of Sines or the Law of Cosines is needed to solve the triangle. (a) \(A\), \(C\), and \(a\) (b) \(a\), \(c\), and \(C\) (c) \(b\), \(c\), and \(A\) (d) \(A\), \(B\), and \(c\) (e) \(b\), \(c\), and \(C\) (f) \(a\), \(b\), and \(c\)

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \(\left(\cos \dfrac{\pi}{4} + i\ \sin \dfrac{\pi}{4}\right)^{12}\)

In Exercises 75-78, find the component form of the sum of \(\mathbf{u}\) and \(\mathbf{v}\) with direction angles \(\mathbf{\theta_u}\) and \(\mathbf{\theta_v}\). Magnitude ||\(\small{\mathbf{u}}\)|| \(= 5\) ||\(\small{\mathbf{v}}\)|| \(= 5\) Angle \(\mathbf{\theta_u} = 0^{\circ}\) \(\mathbf{\theta_v} = 90^{\circ}\)

PROOF Use the Law of Cosines to prove that \(\dfrac{1}{2}bc(1+ \cos A)\ =\ \dfrac{a+b+c}{2} \cdot \dfrac{-a+b+c}{2}\).

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \(\left[2\left(\cos \dfrac{\pi}{2} + i\ \sin \dfrac{\pi}{2}\right)\right]^8\)

BRAKING LOAD A sport utility vehicle with a gross weight of 5400 pounds is parked on a slope of \(10^{\circ}\). Assume that the only force to overcome is the force of gravity. Find the force required to keep the vehicle from rolling down the hill. Find the force perpendicular to the hill.

In Exercises 75-78, find the component form of the sum of \(\mathbf{u}\) and \(\mathbf{v}\) with direction angles \(\mathbf{\theta_u}\) and \(\mathbf{\theta_v}\). Magnitude ||\(\small{\mathbf{u}}\)|| \(= 4\) ||\(\small{\mathbf{v}}\)|| \(= 4\) Angle \(\mathbf{\theta_u} = 60^{\circ}\) \(\mathbf{\theta_v} = 90^{\circ}\)

PROOF Use the Law of Cosines to prove that \(\dfrac{1}{2}bc(1- \cos A)\ =\ \dfrac{a-b+c}{2} \cdot \dfrac{a+b-c}{2}\).

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \([5(\cos\ 3.2 + i\ \sin\ 3.2)]^{4}\)

WORK Determine the work done by a person lifting a 245-newton bag of sugar 3 meters.

In Exercises 75-78, find the component form of the sum of \(\mathbf{u}\) and \(\mathbf{v}\) with direction angles \(\mathbf{\theta_u}\) and \(\mathbf{\theta_v}\). Magnitude ||\(\small{\mathbf{u}}\)|| \(= 20\) ||\(\small{\mathbf{v}}\)|| \(= 50\) Angle \(\mathbf{\theta_u} = 45^{\circ}\) \(\mathbf{\theta_v} = 180^{\circ}\)

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \((\cos\ 0 + i\ \sin\ 0)^{20}\)

In Exercises 75-78, find the component form of the sum of \(\mathbf{u}\) and \(\mathbf{v}\) with direction angles \(\mathbf{\theta_u}\) and \(\mathbf{\theta_v}\). Magnitude ||\(\small{\mathbf{u}}\)|| \(= 50\) ||\(\small{\mathbf{v}}\)|| \(= 30\) Angle \(\mathbf{\theta_u} = 30^{\circ}\) \(\mathbf{\theta_v} = 110^{\circ}\)

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \((3\ -\ 2i)^5\)

WORK A force of 45 pounds exerted at an angle of \(30^{\circ}\) above the horizontal is required to slide a table across a floor (see figure). The table is dragged 20 feet. Determine the work done in sliding the table.

In Exercises 79 and 80, use the Law of Cosines to find the angle \(\alpha\) between the vectors. ( Assume \(\small{0^{\circ} \neq \alpha \neq 190^{\circ}}\). ) \(\small{\mathbf{v} = \mathbf{i} + \mathbf{j}}\), \(\small{\mathbf{w} = 2\mathbf{i} - 2\mathbf{j}}\)

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \((\sqrt{5}\ -\ 4i)^3\)

WORK A tractor pulls a log 800 meters, and the tension in the cable connecting the tractor and log is approximately 15,691 newtons. The direction of the force is \(35^{\circ}\) above the horizontal. Approximate the work done in pulling the log.

In Exercises 79 and 80, use the Law of Cosines to find the angle \(\alpha\) between the vectors. ( Assume \(\small{0^{\circ} \neq \alpha \neq 190^{\circ}}\). ) \(\small{\mathbf{v} = \mathbf{i} + 2\mathbf{j}}\), \(\small{\mathbf{w} = 2\mathbf{i} - \mathbf{j}}\)

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \([3(\cos\ 15^{\circ}\ +\ i\ \sin\ 15^{\circ}]^4\)

RESULTANT FORCE In Exercises 81 and 82, find the angle between the forces given the magnitude of their resultant. (Hint: Write force 1 as a vector in the direction of the positive \(x\)-axis and force 2 as a vector at an angle \(\theta\) with the positive \(x\)-axis.) Force 1 - 45 pounds Force 2 - 60 pounds Resultant Force - 90 pounds

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \(\left[2\left(\cos\ \dfrac{\pi}{8}\ +\ i\ \sin\ \dfrac{\pi}{8}\right)\right]^6\)

VELOCITY A gun with a muzzle velocity of 1200 feet per second is fired at an angle of \(6^{\circ}\) above the horizontal. Find the vertical and horizontal components of the velocity.

Detroit Tigers pitcher Joel Zumaya was recorded throwing a pitch at a velocity of 104 miles per hour. If he threw the pitch at an angle of \(35^{\circ}\) below the horizontal, find the vertical and horizontal components of the velocity. (Source: Damon Lichtenwalner, Baseball Info Solutions)

In Exercises 83-98, (a) use the formula on page 474 to find the indicated roots of the complex number, (b) represent each of the roots graphically, and (c) write each of the root sin standard form. Cube roots of \(8 \left(\cos \dfrac{2\pi}{3}\ +\ i\ \sin \dfrac{2\pi}{3} \right)\)

PROGRAMMING Given vectors \(\mathbf{u}\) and \(\mathbf{v}\) in component form, write a program for your graphing utility in which the output is the component form of the projection of \(\mathbf{u}\) onto \(\mathbf{v}\).

RESULTANT FORCE Three forces with magnitudes of 75 pounds, 100 pounds, and 125 pounds act on an object at angles of \(30^{\circ}\), \(45^{\circ}\), and \(120^{\circ}\) respectively, with the positive \(x\)-axis. Find the direction and magnitude of the resultant of these forces.

TRUE OR FALSE? In Exercises 87 and 88, determine whether the statement is true or false. Justify your answer. A sliding door moves along the line of vector \(\overset{\rightharpoonup} {PQ}\). If a force is applied to the door along a vector that isorthogonal to \(\overset{\rightharpoonup} {PQ}\), then no work is done.

RESULTANT FORCE Three forces with magnitudes of 70 pounds, 40 pounds, and 60 pounds act on an object at angles of \(-30^{\circ}\), \(45^{\circ}\), and \(135^{\circ}\) respectively, with the positive \(x\)-axis. Find the direction and magnitude of the resultant of these forces.

PROOF Use vectors to prove that the diagonals of a rhombus are perpendicular.

THINK ABOUT IT What can be said about the vectors \(\mathbf{u}\) and \(\mathbf{v}\) under each condition? (a) The projection of \(\mathbf{u}\) onto \(\mathbf{v}\) equals \(\mathbf{u}\). (b) The projection of \(\mathbf{u}\) onto \(\mathbf{v}\) equals \(\mathbf{0}\).