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524 7 Additional Topics in Trigonometry miles above the surface of the Earth (see figure). (A satellite in geostationary orbit remains stationary above a fixed point on the surface of the Earth.) Radio signals are sent from the tracking station by way of the satellites to the shuttle, and vice versa. This system allows the tracking station to be in contact with the shuttle over most of the Earth’s surface. How far to the nearest 100 miles is one of the geostationary satellites from the White Sands tracking station W? The radius of the Earth is 3,964 miles. S ★ 48. Space Science. A satellite S, in circular orbit around the Earth, is sighted by a tracking station T (see figure). The distance TS is determined by radar to be 1,034 miles, and the angle of elevation above the horizon is 32.4°. How high is the satellite above the Earth at the time of the sighting? The radius of the Earth is 3,964 miles. S T Horizon S R W Earth C C SECTION 7-3 Geometric Vectors • • • • Geometric Vectors and Vector Addition Velocity Vectors Force Vectors Resolution of Vectors into Vector Components Many physical quantities, such as length, area, or volume, can be completely specified by a single real number. Other quantities, such as directed distances, velocities, and forces, require for their complete specification both a magnitude and a direction. The former are often called scalar quantities, and the latter are called vector quantities. In this section we limit our discussion to the intuitive idea of geometric vectors in a plane. In Section 7-4 we introduce algebraic vectors, a first step in the generalization of a concept that has far-reaching consequences. Vectors are widely used in many areas of science and engineering. • Geometric Vectors and Vector Addition P v O , or v. FIGURE 1 Vector OP A line segment to which a direction has been assigned is called a directed line segment. A geometric vector is a directed line segment and is represented by an arrow (see Fig. 1). A vector with an initial point O and a terminal point P (the end with . Vectors are also denoted by a boldface letter, such the arrowhead) is denoted by OP as v. Since it is difficult to write boldface on paper, we suggest that you use an arrow over a single letter, such as v , when you want the letter to denote a vector. , denoted by OP , v or v, is the length of the The magnitude of the vector OP directed line segment. Two vectors have the same direction if they are parallel and point in the same direction. Two vectors have opposite direction if they are parallel and point in opposite directions. The zero vector, denoted by 0 or 0, has a magni- 7-3 u v v u FIGURE 2 Vector addition: tail-totip rule. v u v u FIGURE 3 Vector addition: parallelogram rule. EXPLORE-DISCUSS 1 Geometric Vectors 525 tude of zero and an arbitrary direction. Two vectors are equal if they have the same magnitude and direction. Thus, a vector may be translated from one location to another as long as the magnitude and direction do not change. The sum of two vectors u and v can be defined using the tail-to-tip rule: Translate v so that its tail end (initial point) is at the tip end (terminal point) of u. Then, the vector from the tail end of u to the tip end of v is the sum, denoted by u v, of the vectors u and v (see Fig. 2). The sum of two nonparallel vectors also can be defined using the parallelogram rule: The sum of two nonparallel vectors u and v is the diagonal of the parallelogram formed using u and v as adjacent sides (see Fig. 3). If u and v are parallel, use the tail-to-tip rule. Both rules give the same sum. The choice of which rule to use depends on the situation and what seems most natural. The vector u v is also called the resultant of the two vectors u and v, and u and v are called vector components of u v. It is useful to observe that vector addition is commutative and associative. That is, u v v u and u (v w) (u v) w. If a, b and c represent three arbitrary geometric vectors, illustrate using either definition of vector addition that: 1. a b b a 2. a (b c) (a b) c • Velocity Vectors A vector that represents the direction and speed of an object in motion is called a velocity vector. Problems involving objects in motion often can be analyzed using vector methods. Many of these problems involve the use of a navigational compass, which is marked clockwise in degrees starting at north as indicated in Figure 4. FIGURE 4 N, 0 W, 270 90, E S, 180 EXAMPLE 1 Apparent and Actual Velocity An airplane has a compass heading (the direction the plane is pointing) of 85° and an airspeed (relative to the air) of 140 miles per hour. The wind is blowing from north to south at 66 miles per hour. The velocity of a plane relative to the air is called apparent velocity, and the velocity relative to the ground is called resultant, or actual velocity. The resultant velocity is the vector sum of the apparent velocity and the wind velocity. Find the resultant velocity; that is, find the actual speed and 526 7 Additional Topics in Trigonometry direction of the airplane relative to the ground. Directions are given to the nearest degree and magnitudes to 2 significant digits. Solution Geometric vectors [Fig. 5 (a)] are used to represent the apparent velocity vector and the wind velocity vector. Add the two vectors using the tail-to-tip method of addition of vectors to obtain the resultant (actual) velocity vector [Fig. 5 (b)]. From the vector diagram [Fig. 5 (b)], we obtain the triangle in Figure 6 and solve for , c, and . FIGURE 5 N N Apparent velocity 85 Wind velocity Apparent velocity 85 180 Actual velocity (a) FIGURE 6 Wind velocity (b) 140 66 c Solve for Since the wind velocity vector is parallel to the north–south line, 85° [alternate interior angles of two parallel lines cut by a transversal are equal—see Fig. 5 (b)]. Solve for c Use the law of cosines: c2 a2 b2 2ab cos c a2 b2 2ab cos 662 1402 2(66)(140) cos 85° 150 miles per hour Solve for Speed relative to the ground. Use the law of sines: sin sin a c a sinb sin 85 66 150 26° sin1 sin1 Actual heading 85° 85° 26° 111° Thus, the magnitude and direction of the resultant velocity vector are 150 miles per hour and 111°, respectively. That is, the plane, relative to the ground, is traveling at 150 miles per hour in a direction of 111°. 7-3 527 Geometric Vectors Matched Problem 1 A river is flowing southwest (225°) at 3.0 miles per hour. A boat crosses the river with a compass heading of 90°. If the speedometer on the boat reads 5.0 miles per hour (the boat’s speed relative to the water), what is the resultant velocity? That is, what is the boat’s actual speed and direction relative to the ground? Directions are to the nearest degree, and magnitudes are to 2 significant digits. • Force Vectors A vector that represents the direction and magnitude of an applied force is called a force vector. If an object is subjected to two forces, then the sum of these two forces, the resultant force, is a single force. If the resultant force replaced the original two forces, it would act on the object in the same way as the two original forces taken together. In physics it is shown that the resultant force vector can be obtained using vector addition to add the two individual force vectors. It seems natural to use the parallelogram rule for adding force vectors, as is illustrated in the next example. EXAMPLE 2 Finding the Resultant Force Two forces of 30 and 70 pounds act on a point in a plane. If the angle between the force vectors is 40°, what are the magnitude and direction (relative to the 70-pound force) of the resultant force? The magnitudes of the forces are to 2 significant digits and the angles to the nearest degree. Solution We start with a diagram (Fig. 7), letting geometric vectors represent the various forces: FIGURE 7 A B R 30 pounds 40 O 70 pounds C Because adjacent angles in a parallelogram are supplementary, the measure of angle OCB 180° 40° 140°. We can now find the magnitude of the resultant vector R using the law of cosines (see Fig. 8): R R2 302 702 2(30)(70) cos 140° 95 pounds 140 R 302 702 2(30)(70) cos 140° 70 FIGURE 8 To find , the direction of R, we use the law of sines (see Fig. 9): sin sin 140° 30 95 sin 30 sin 140° 95 95 140 70 FIGURE 9 30 30 528 7 Additional Topics in Trigonometry sin1 30 sin95140° 12° Thus, the two given forces are equivalent to a single force of 95 pounds in the direction of 12° (relative to the 70-pound force). Matched Problem 2 • Resolution of Vectors into Vector Components EXAMPLE 3 Repeat Example 2 using an angle of 100° between the two forces. Instead of adding vectors, many problems require the resolution of vectors into components. As we indicated earlier, whenever a vector is expressed as the sum or resultant of two vectors, the two vectors are called vector components of the given vector. Example 3 illustrates an application of the process of resolving a vector into vector components. Resolving a Force Vector into Components A car weighing 3,210 pounds is on a driveway inclined 20.2° to the horizontal. Neglecting friction, find the magnitude of the force parallel to the driveway that will keep the car from rolling down the hill. Solution We start by drawing a vector diagram (Fig. 10): FIGURE 10 y ewa Driv A D 20.2 C B 3,210 pounds acts in a downward direction and represents the weight of The force vector DB DA , where DC is the perpendicular component of DB the car. Note that DB DC relative to the driveway and DA is the parallel component of DB relative to the driveway. To keep the car at D from rolling down the hill, we need a force with the mag but oppositely directed. To find DA , we first observe that ABD nitude of DA 20.2°. This is true because ABD and the driveway angle have the same complement, ADB. 7-3 sin 20.2° Geometric Vectors 529 DA 3,210 3,210 sin 20.2° DA 1,110 pounds in Example 3. Find the magnitude of the perpendicular component of DB Matched Problem 3 Answers to Matched Problems 1. Resultant velocity: magnitude 3.6 mph, direction 126° 2. R 71 lb, 25° 3,010 lb 3. DC EXERCISE 7-3 7. u v 390 miles per hour, 6° Express all angle measures in decimal degrees. 8. u v 75 miles per hour, 81° A Problems 1–8 refer to figures (a) and (b) showing vector addition for vectors u and v at right angles to each other. u v v B Problems 9–16 refer to figures (c) and (d) showing vector addition for vectors u and v. u v v u v u v u Tail-to-tip rule Parallelogram rule u (a) (b) In Problems 1–4, find u v and , given u and v in figures (a) and (b). 1. u 30 miles per hour, v 72 miles per hour 2. u 216 miles per hour, v 63 miles per hour u v v u Tail-to-tip rule Parallelogram rule (c) In Problems 9–12, find u v and , given u, v, and in figures (c) and (d). 9. u 66 grams, v 22 grams, 68° 3. u 29 kilograms, v 29 kilograms 10. u 120 grams, v 84 grams, 44° 4. u 78 kilograms, v 45 kilograms 11. u 21 knots, v 3.2 knots, 53° In Problems 5–8, find u and v, the magnitudes of the horizontal and vertical components of u v, given u v and in figures (a) and (b). (d) 12. u 8.0 knots, v 2.0 knots, 64° In Problems 13–16, find u and v, given u v, and in figures (c) and (d). 5. u v 24 pounds, 60° 13. u v 14 kilograms, 25°, 79° 6. u v 48 pounds, 45° 14. u v 33 kilograms, 17°, 43° 530 7 Additional Topics in Trigonometry 15. u v 223 miles per hour, 42.3°, 69.4° river’s current is 2.5 knots, what compass heading should be maintained while crossing the river? What is the actual speed of the boat relative to the land? 16. u v 437 miles per hour, 17.8°, 50.5° ★ 28. Navigation. An airplane can cruise at 255 miles per hour in still air. If a steady wind of 46.0 miles per hour is blowing from the west, what compass heading should the pilot fly in order for the course of the plane relative to the ground to be north (0°)? Compute the ground speed for this course. ★ 29. Resultant Force. A large ship has gone aground in a harbor and two tugs, with cables attached, attempt to pull it free. If one tug pulls with a compass course of 52° and a force of 2,300 pounds and a second tug pulls with a compass course of 97° and a force of 1,900 pounds, what is the compass direction and the magnitude of the resultant force? ★ 30. Resultant Force. Repeat Problem 29 if one tug pulls with a compass direction of 161° and a force of 2,900 kilograms and a second tug pulls with a compass direction of 192° and a force of 3,600 kilograms. C In Problems 17–24, determine whether the statement is true or false. If true, explain why. If false, give a counterexample. 17. The zero vector is perpendicular to every vector. 18. The zero vector is parallel to every vector. 19. Vectors having the same magnitude are equal. 20. Equal vectors have the same magnitude. 21. Equal vectors have the same direction. 22. Perpendicular vectors have the opposite direction. 23. The magnitude of every vector is positive. 24. The magnitude of u v is greater than u. 31. Resolution of Forces. An automobile weighing 4,050 pounds is standing on a driveway inclined 5.5° with the horizontal. (A) Find the magnitude of the force parallel to the driveway necessary to keep the car from rolling down the hill. (B) Find the magnitude of the force perpendicular to the driveway. APPLICATIONS In navigation problems, refer to the figure of a navigational compass below: 32. Resolution of Forces. Repeat Problem 31 for a car weighing 2,500 pounds parked on a hill inclined at 15° to the horizontal. N, 0 ★ W, 270 90, E 33. Resolution of Forces. If two weights are fastened together and placed on inclined planes as shown in the figure, neglecting friction, which way will they slide? S, 180 Navigational compass 110 ds n pou In Problems 25–28, assume the north, east, south, and west directions are exact. 25. Navigation. An airplane is flying with a compass heading of 285° and an airspeed of 230 miles per hour. A steady wind of 35 miles per hour is blowing in the direction of 260°. What is the plane’s actual velocity; that is, what is its speed and direction relative to the ground? 25 ★ 27. Navigation. Two docks are directly opposite each other on a southward-flowing river. A boat pilot wishes to go in a straight line from the east dock to the west dock in a ferryboat with a cruising speed in still water of 8.0 knots. If the 35 34. Resolution of Forces. If two weights are fastened together and placed on inclined planes as indicated in the figure, neglecting friction, which way will they slide? 26. Navigation. A powerboat crossing a wide river has a compass heading of 25° and speed relative to the water of 15 miles per hour. The river is flowing in the direction of 135° at 3.9 miles per hour. What is the boat’s actual velocity; that is, what is its speed and direction relative to the ground? ★ po 85 un ds 41 ds un po 31 po 31 un ds 41