Chapter 7

Find the magnitude of each of the following vectors. $$\langle-5,6\rangle$$

Each of the following problems refers to triangle \(A B C\). In each case, find the area of the triangle. Round to three significant digits. \(a=4.38 \mathrm{ft}, b=3.79 \mathrm{ft}, c=5.22 \mathrm{ft}\)

Current A ship is headed due north at a constant 16 miles per hour. Because of the ocean current, the true course of the ship is \(15^{\circ}\). If the currents are a constant 14 miles per hour, in what direction are the currents running?

$$ \text { Solve each of the following triangles. } $$ $$ \text { Use the law of cosines to show that, if } a^{2}=b^{2}+c^{2} \text {, then } A=90^{\circ} \text {. } $$

Find the angle \(\theta\) between the given vectors to the nearest tenth of a degree. \(\mathrm{U}=4 \mathrm{i}+5 \mathrm{j}, \mathrm{V}=7 \mathrm{i}-4 \mathrm{j}\)

Find the magnitude of each of the following vectors. $$\langle-9,-2\rangle$$

Each of the following problems refers to triangle \(A B C\). In each case, find the area of the triangle. Round to three significant digits. \(a=8.32 \mathrm{ft}, b=12.36 \mathrm{ft}, c=5.34 \mathrm{ft}\)

$$ \text { Solve each of the following triangles. } $$ Geometry The diagonals of a parallelogram are 14 meters and 16 meters and intersect at an angle of \(60^{\circ}\). Find the length of the longer side.

Find the angle \(\theta\) between the given vectors to the nearest tenth of a degree. \(\mathbf{U}=13 \mathbf{i}-8 \mathbf{j}, \mathbf{V}=2 \mathbf{i}+11 \mathbf{j}\)

Find the magnitude of each of the following vectors. $$\langle 0,5\rangle$$

Geometry and Area Find the area of a parallelogram if the angle between two of the sides is \(120^{\circ}\) and the two sides are 15 inches and 12 inches in length.

Ground Speed A ship headed due east is moving through the water at a constant speed of 12 miles per hour. However, the true course of the ship is \(60^{\circ}\). If the currents are a constant 6 miles per hour, what is the ground speed of the ship?

$$ \text { Solve each of the following triangles. } $$ Geometry The diagonals of a parallelogram are 56 inches and 34 inches and intersect at an angle of \(130^{\circ}\). Find the length of the shorter side.

Find the angle \(\theta\) between the given vectors to the nearest tenth of a degree. \(\mathbf{U}=11 \mathbf{i}+7 \mathbf{j}, \mathbf{V}=-14 \mathbf{i}+6 \mathbf{j}\)

Find the magnitude of each of the following vectors. $$\langle-7,0\rangle$$

True Course A plane headed due east is traveling with an airspeed of 190 miles per hour. The wind currents are moving with constant speed in the direction \(240^{\circ}\). If the ground speed of the plane is 95 miles per hour, what is its true course?

$$ \text { Solve each of the following triangles. } $$ Distance Between Two Planes Two planes leave an airport at the same time. Their speeds are 130 miles per hour and 150 miles per hour, and the angle between their courses is \(36^{\circ}\). How far apart are they after \(1.5\) hours?

Show that each pair of vectors is perpendicular. \(\mathbf{i}+\mathbf{j}\) and \(\mathrm{i}-\mathrm{j}\) \(i\) and \(j\)

Find the magnitude of each of the following vectors. $$\mathbf{U}=5 \mathbf{i}+12 \mathbf{j}$$

Leaning Windmill After a wind storm, a farmer notices that his 32 -foot windmill may be leaning, but he is not sure. From a point on the ground 31 feet from the base of the windmill, he finds that the angle of elevation to the top of the windmill is \(48^{\circ}\). Is the windmill leaning? If so, what is the acute angle the windmill makes with the ground?

$$ \text { Solve each of the following triangles. } $$ Distance Between Two Ships Two ships leave a harbor entrance at the same time. The first ship is traveling at a constant 18 miles per hour, while the second is traveling at a constant 22 miles per hour. If the angle between their courses is \(123^{\circ}\), how far apart are they after 30 minutes?

Show that each pair of vectors is perpendicular. \(-\mathbf{i}\) and \(\mathbf{j}\)

Find the magnitude of each of the following vectors. $$\mathbf{U}=15 \mathbf{i}-8 \mathbf{j}$$

Distance \(A\) boy is riding his motorcycle on a road that runs east and west. He leaves the road at a service station and rides \(5.25\) miles in the direction \(\mathrm{N} \mathrm{} 15.5^{\circ} \mathrm{E}\). Then he turns to his right and rides \(6.50\) miles back to the road, where his motorcycle breaks down. How far will he have to walk to get back to the service station?

Angle of Elevation A woman entering an outside glass elevator on the ground floor of a hotel glances up to the top of the building across the street and notices that the angle of elevation is \(48^{\circ}\). She rides the elevator up three floors ( 60 feet) and finds that the angle of elevation to the top of the building across the street is \(32^{\circ}\). How tall is the building across the street? (Round to the nearest foot.)

Show that each pair of vectors is perpendicular. \(2 \mathbf{i}+\mathbf{j}\) and \(\mathbf{i}-2 \mathbf{j}\)

Find the magnitude of each of the following vectors. $$\mathbf{W}=-\mathbf{i}-2 \mathbf{j}$$

We know from this section that the area of any triangle \(A B C\) is given by Arca \(=\frac{1}{2} b c \sin A=\frac{1}{2} a c \sin B=\frac{1}{2} a b \sin C\) Use this fact to derive the law of sines.

A sailboat set a course of \(\mathrm{N} 25^{\circ} \mathrm{E}\) from a small port along a shoreline that runs north and south. Sometime later the boat overturned and the crew sent out a distress call. They estimated that they were 12 miles away from the nearest harbor, which is 28 miles north of the port they had set sail from. If a rescue team leaves from the harbor, find all possible courses the team must follow in order to reach the overturned sailboat.

Angle of Elevation A 155 -foot antenna is on top of a tall building. From a point on the ground, the angle of elevation to the top of the antenna is \(28.5^{\circ}\), while the angle of elevation to the bottom of the antenna from the same point is \(23.5^{\circ}\). How tall is the building?

Show that each pair of vectors is perpendicular. \(-4 \mathbf{i}-3 \mathbf{j}\) and \(6 \mathbf{i}-8 \mathbf{j}\)

Find the magnitude of each of the following vectors. $$\mathbf{W}=-3 \mathbf{i}+\mathbf{j}$$

Draw vectors representing the course of a ship that travels $$ 25 \text { miles on a course with heading } 225^{\circ} $$

For each vector, find \(\frac{1}{2} \mathbf{V},-\mathbf{V}\), and \(4 \mathbf{V}\). $$\mathbf{V}=\langle-3,7\rangle$$

The problems that follow review material we covered in Section 6.2. Find all solutions in the interval \(0^{\circ} \leq \theta<360^{\circ}\). If rounding is necessary, round to the nearest tenth of a degree. $$4 \sin \theta-\csc \theta=0$$

Draw vectors representing the course of a ship that travels $$ 25 \text { miles on a course with heading } 135^{\circ} $$

Show that each pair of vectors is perpendicular. Find the value of \(a\) so that vectors \(\mathbf{U}=a \mathbf{i}+6 \mathbf{j}\) and \(\mathbf{V}=9 \mathbf{i}+12 \mathbf{j}\) are perpendicular

For each vector, find \(\frac{1}{2} \mathbf{V},-\mathbf{V}\), and \(4 \mathbf{V}\). $$V=\langle-2,5\rangle$$

Distance to a Ship A ship is anchored off a long straight shoreline that runs north and south. From two observation points 18 miles apart on shore, the bearings of the ship are \(\mathrm{N} 31^{\circ} \mathrm{E}\) and \(\mathrm{S} 53^{\circ} \mathrm{E}\). What is the distance from the ship to each of the observation points?

For Problems 37 through 42, use your knowledge of bearing, heading, and true course to sketch a diagram that will help you solve each problem. Heading and Distance Two planes take off at the same time from an airport. The first plane is flying at 246 miles per hour on a course of \(135.0^{\circ}\). The second plane is flying in the direction \(175.0^{\circ}\) at 357 miles per hour. Assuming there are no wind currents blowing, how far apart are they after 2 hours?

Show that each pair of vectors is perpendicular. In general, show that the vectors \(\mathbf{V}=a \mathbf{i}+b \mathbf{j}\) and \(\mathbf{W}=-b \mathbf{i}+a \mathbf{j}\) are always perpendicular. Assume \(a\) and \(b\) are not both equal to zero.

For each vector, find \(\frac{1}{2} \mathbf{V},-\mathbf{V}\), and \(4 \mathbf{V}\). $$V=2 \mathbf{i}+4 \mathbf{j}$$

The problems that follow review material we covered in Section 6.2. Find all solutions in the interval \(0^{\circ} \leq \theta<360^{\circ}\). If rounding is necessary, round to the nearest tenth of a degree. $$2 \cos \theta-\sin 2 \theta=0$$

Distance to a Rocket Tom and Fred are \(3.5\) miles apart watching a rocket being launched from Vandenberg Air Force Base. Tom estimates the bearing of the rocket from his position to be \(S 75^{\circ} \mathrm{W}\), while Fred estimates that the bearing of the rocket from his position is N \(65^{\circ} \mathrm{W}\). If Fred is due south of Tom, how far is each of them from the rocket?

For Problems 37 through 42, use your knowledge of bearing, heading, and true course to sketch a diagram that will help you solve each problem. Bearing and Distance Two ships leave the harbor at the same time. One ship is traveling at 15 miles per hour on a course with a bearing of \(\mathrm{S} \mathrm{} 13^{\circ} \mathrm{W}\), while the other is traveling at 11 miles per hour on a course with a bearing of \(\mathrm{N} 75^{\circ} \mathrm{E}\). How far apart are they after 3 hours?

Find the work performed when the given force \(\mathbf{F}\) is applied to an object, whose resulting motion is represented by the displacement vector \(d\). Assume the force is in pounds and the displacement is measured in feet. \(\mathbf{F}=22 \mathbf{i}+9 \mathbf{j}, \mathbf{d}=30 \mathbf{i}+4 \mathbf{j}\)

For each vector, find \(\frac{1}{2} \mathbf{V},-\mathbf{V}\), and \(4 \mathbf{V}\). $$V=-\sqrt{3} \mathbf{i}-\mathbf{j}$$

The problems that follow review material we covered in Section 6.2. Find all solutions in the interval \(0^{\circ} \leq \theta<360^{\circ}\). If rounding is necessary, round to the nearest tenth of a degree. $$\cos 2 \theta+3 \cos \theta-2=0$$

For Problems 37 through 42, use your knowledge of bearing, heading, and true course to sketch a diagram that will help you solve each problem. True Course and Speed A plane is flying with an airspeed of 160 miles per hour and heading of \(150^{\circ}\). The wind currents are running at 35 miles per hour at \(165^{\circ}\) clockwise from due north. Use vectors to find the true course and ground speed of the plane.

Find the work performed when the given force \(\mathbf{F}\) is applied to an object, whose resulting motion is represented by the displacement vector \(d\). Assume the force is in pounds and the displacement is measured in feet. \(\mathbf{F}=45 \mathbf{i}-12 \mathbf{j}, \mathbf{d}=170 \mathbf{i}+15 \mathbf{j}\)