1 7-4 lgebraic Vectors 531 SECTIN 7-4 lgebraic Vectors From Geometric Vectors to lgebraic Vectors Vector ddition and Scalar Multiplication Unit Vectors lgebraic Properties Static Equilibrium Geometric vectors in a plane are readil generalized to three-dimensional space. However, to generalize vectors further to higher-dimensional abstract spaces, it is essential to define the vector concept algebraicall. This is done in such a wa that the geometric vectors become special cases of the more general algebraic vectors. lgebraic vectors have man advantages over geometric vectors. ne advantage will become apparent when we consider static equilibrium problems at the end of the section. The development of algebraic vectors in this book is introductor in nature and is restricted to the plane. Further stud of vectors in three- and higher-dimensional spaces is reserved for more advanced mathematical courses. From Geometric Vectors to lgebraic Vectors The transition from geometric vectors to algebraic vectors is begun b placing geometric vectors in a rectangular coordinate sstem. geometric vector in a rectangular coordinate sstem translated so that its initial point is at the origin is said to be in standard position. The vector P such that P is said to be the standard vector for (see Fig. 1). Note that the vector P in Figure 1 is the standard vector for infinitel man vectors all vectors with the same magnitude and direction as P. P Standard vector FIGURE 1 P is the standard vector for. EXPLRE-DISCUSS 1 () In a cop of Figure 1, draw in three other vectors having P as their standard vector. () If the tail of a vector is at point ( 3, 2) and its tip is at (6, 4), discuss how ou would find the coordinates of P so that P is the standard vector for. Given the coordinates of the endpoints of a geometric vector in a rectangular coordinate sstem, how do we find its corresponding standard vector? The process is not difficult. The coordinates of the initial point,, of P are alwas (0, 0). Thus, we have onl to find the coordinates of P, the terminal point of P. The coordinates of P are given b ( p, p ) ( b a, b a ) (1)
2 532 7 dditional Topics in Trigonometr where the coordinates of are ( a, a ) and the coordinates of are ( b, b ). Eample 1 illustrates the use of equation (1). EXMPLE 1 Finding a Standard Vector for a Given Vector Standard vector FIGURE 2 Solution (3, 4) (7, 1) P(4, 5) Given the geometric vector with initial point (3, 4) and terminal point (7, 1), find the standard vector P for. That is, find the coordinates of the point P such that P. The coordinates of P are given b ( p, p ) ( b a, b a ) (7 3, 1 4) (4, 5) Note in Figure 2 that if we start at, then move to the right 4 units and down 5 units, we will be at. If we start at the origin, then move to the right 4 units and down 5 units, we will be at P. Matched Problem 1 Given the geometric vector with initial point (8, 3) and terminal point (4, 5), find the standard vector P for. P(a, b) v a, b FIGURE 3 lgebraic vector a, b associated with a geometric vector P. The preceding discussion suggests another wa of looking at vectors. Since, given an geometric vector in a rectangular coordinate sstem, there alwas eists a point P( p, p ) such that P, the point P( p, p ) completel specifies the vector, ecept for its position. nd we are not concerned about its position because we are free to translate anwhere we please. Conversel, given an point P( p, p ) in a rectangular coordinate sstem, the directed line segment joining to P forms the geometric vector P. This leads us to define an algebraic vector as an ordered pair of real numbers. To avoid confusing a point (a, b) with a vector (a, b), we use a, b to represent an algebraic vector. Geometricall, the algebraic vector a, b corresponds to the standard (geometric) vector P with terminal point P(a, b) and initial point (0, 0), as illustrated in Figure 3. The real numbers a and b are scalar components of the vector a, b. The word scalar means real number and is often used in the contet of vectors where one refers to scalar quantities as opposed to vector quantities. Thus, we talk about scalar components and vector components of a given vector. The words scalar and vector are often dropped if the meaning of component is clear from the contet. Two vectors u a, b and v c, d are said to be equal if their corresponding components are equal, that is, if a c and b d. The zero vector is denoted b 0 0, 0. Geometric vectors are limited to spaces we can visualize, that is, to two- and three-dimensional spaces. lgebraic vectors do not have these restrictions. The following are algebraic vectors from two-, three-, four-, and five-dimensional spaces:
3 7-4 lgebraic Vectors 533 2, 5 3, 0, 8 5, 1, 1, 2 1, 0, 1, 3, 4 s we said earlier, the discussion in this book is limited to algebraic vectors in a twodimensional space, which represents a plane. We now define the magnitude of an algebraic vector: DEFINITIN 1 Magnitude of v a, b The magnitude, or norm, of a vector v a, b is denoted b v and is given b P(a, b) v a 2 b 2 v a 2 b 2 FIGURE 4 Magnitude of vector a, b geometricall interpreted. Geometricall, a 2 b 2 is the length of the standard geometric vector P associated with the algebraic vector a, b (see Fig. 4). The definition of magnitude is readil generalized to higher-dimensional vector spaces. For eample, if v a, b, c, d, then the magnitude, or norm, is given b a 2 b 2 c 2 d 2. ut now we are not able to interpret the result in terms of geometric vectors. EXMPLE 2 Finding the Magnitude of a Vector Find the magnitude of the vector v 3, 5. Solution v 3 2 ( 5) 2 34 Matched Problem 2 Find the magnitude of the vector v 2, 4. Vector ddition and Scalar Multiplication To add two algebraic vectors, add the corresponding components as indicated in the following definition of addition: DEFINITIN 2 Vector ddition If u a, b and v c, d, then u v a c, b d The definition of addition of algebraic vectors is consistent with the parallelogram and tail-to-tip definitions for adding geometric vectors given in Section 7-3 (see Eplore-Discuss 2).
4 534 7 dditional Topics in Trigonometr EXPLRE-DISCUSS 2 If u 3, 2, v 7, 3, then u v 3 7, 2 3 4, 5. Locate u, v, and u v in a rectangular coordinate sstem and interpret geometricall in terms of the parallelogram and tail-to-tip rules discussed in the last section. To multipl a vector b a scalar (a real number) multipl each component b the scalar: DEFINITIN 3 Scalar Multiplication If u a, b and k is a scalar, then ku k a, b ka, kb v v 2v 0.5v Geometricall, if a vector v is multiplied b a scalar k, the magnitude of the vector v is multiplied b k. If k is positive, then kv has the same direction as v. If k is negative, then kv has the opposite direction as v. These relationships are illustrated in Figure 5. FIGURE 5 Scalar multiplication geometricall interpreted. EXMPLE 3 Vector ddition and Scalar Multiplication Let u 4, 3, v 2, 3, and w 0, 5, find: () u v () 2u (C) 2u 3v (D) 3u 2v w Solutions () u v 4, 3 2, 3 6, 0 () 2u 2 4, 3 8, 6 (C) 2u 3v 2 4, 3 3 2, 3 8, 6 6, 9 2, 15 (D) 3u 2v w 3 4, 3 2 2, 3 0, 5 12, 9 4, 6 0, 5 16, 2 Matched Problem 3 Let u 5, 3, v 4, 6, and w 2, 0, find: () u v () 3u (C) 3u 2v (D) 2u v 3w
5 7-4 lgebraic Vectors 535 Unit Vectors If v 1, then v is called a unit vector. unit vector can be formed from an arbitrar nonzero vector as follows: Unit Vector with the Same Direction as v If v is a nonzero vector, then u 1 v v is a unit vector with the same direction as v. EXMPLE 4 Finding a Unit Vector with the Same Direction as a Given Vector Given a vector v 1, 2, find a unit vector u with the same direction as v. Solution Check v 1 2 ( 2) 2 5 u 1 v v 1 1, , 5 2 u nd we see that u is a unit vector with the same direction as v. Matched Problem 4 Given a vector v 3, 1, find a unit vector u with the same direction as v. We now define two ver important unit vectors, the i and j unit vectors. The i and j Unit Vectors 1 j i 1, 0 j 0, 1 0 i 1
6 536 7 dditional Topics in Trigonometr Wh are the i and j unit vectors so important? ne of the reasons is that an vector v a, b can be epressed as a linear combination of those two vectors; that is, as ai bj. v a, b a, 0 0, b a 1, 0 b 0, 1 ai bj EXMPLE 5 Epressing a Vector in Terms of the i and j Vectors Epress each vector as a linear combination of the i and j unit vectors. () 2, 4 () 2, 0 (C) 0, 7 Solutions () 2, 4 2i 4j () 2, 0 2i 0j 2i (C) 0, 7 0i 7j 7j Matched Problem 5 Epress each vector as a linear combination of the i and j unit vectors. () 5, 3 () 9, 0 (C) 0, 6 lgebraic Properties Vector addition and scalar multiplication possess algebraic properties similar to the real numbers. These properties enable us to manipulate smbols representing vectors and scalars in much the same wa we manipulate smbols that represent real numbers in algebra. These properties are listed below for convenient reference. lgebraic Properties of Vectors. ddition Properties. For all vectors u, v, and w: 1. u v v u Commutative Propert 2. u (v w) (u v) w ssociative Propert 3. u 0 0 u u dditive Identit 4. u ( u) ( u) u 0 dditive Inverse. Scalar Multiplication Properties. For all vectors u and v and all scalars m and n: 1. m(nu) (mn)u ssociative Propert 2. m(u v) mu mv Distributive Propert 3. (m n)u mu nu Distributive Propert 4. 1u u Multiplicative Identit
7 7-4 lgebraic Vectors 537 EXMPLE 6 lgebraic perations on Vectors Epressed in Terms of the i and j Vectors For u i 2j and v 5i 2j, compute each of the following: () u v () u v (C) 2u 3v Solutions () u v (i 2j) (5i 2j) i 2j 5i 2j 6i 0j 6i () u v (i 2j) (5i 2j) i 2j 5i 2j 4i 4j (C) 2u 3v 2(i 2j) 3(5i 2j) 2i 4j 15i 6j 17i 2j Matched Problem 6 For u 2i j and v 4i 5j, compute each of the following: () u v () u v (C) 3u 2v Static Equilibrium lgebraic vectors can be used to solve man tpes of problems in phsics and engineering. We complete this section b considering a few problems involving static equilibrium. Fundamental to our approach are two basic principles regarding forces and objects subject to these forces: Conditions for Static Equilibrium 1. n object at rest is said to be in static equilibrium. 2. For an object located at the origin in a rectangular coordinate sstem to remain in static equilibrium, at rest, it is necessar that the sum of all the force vectors acting on the object be the zero vector. Eample 7 shows how some important phsics/engineering problems can be solved using algebraic vectors and the conditions for static equilibrium. It is assumed that ou know how to solve a sstem of two equations with two variables. In case ou need a reminder, procedures are reviewed in Section 1-2. EXMPLE 7 Tension in Cables cable car, used to ferr people and supplies across a river, weighs 2,500 pounds full loaded. The car stops when partwa across and deflects the cable relative to the
8 538 7 dditional Topics in Trigonometr horizontal, as indicated in Figure 6. What is the tension in each part of the cable running to each tower? FIGURE ,500 pounds River Solution Step 1. Draw a force diagram with all force vectors in standard position at the origin (Fig. 7). The objective is to find u and v. Step 2. Write each force vector in terms of the i and j unit vectors: v 7 u 15 w w 2,500 pounds FIGURE 7 u u (cos 7 )i u (sin 7 )j v v ( cos 15 )i v (sin 15 )j w 2,500j Step 3. For the sstem to be in static equilibrium, the sum of the force vectors must be the zero vector. That is, u v w 0 Replacing vectors u, v, and w from step 2, we obtain [ u (cos 7 )i u (sin 7 )j] [ v ( cos 15 )i v (sin 15 )j] 2,500j 0i 0j which on combining i and j vectors becomes [ u (cos 7 ) v ( cos 15 )]i [ u (sin 7 ) v (sin 15 ) 2,500]j 0i 0j Since two vectors are equal if and onl if their corresponding components are equal, we are led to the following sstem of two equations in the two variables u and v : (cos 7 ) u ( cos 15 ) v 0 (sin 7 ) u (sin 15 ) v 2,500 0 Solving this sstem b standard methods, we find that u 6,400 pounds and v 6,600 pounds Did ou epect that the tension in each part of the cable is more than the weight hanging from the cable?
9 7-4 lgebraic Vectors 539 Matched Problem 7 Repeat Eample 7 with 15 replaced with 13, 7 replaced with 9, and the 2,500 pounds replaced with 1,900 pounds. nswers to Matched Problems 1. P( 4, 8) () 1, 3 () 15, 9 (C) 23, 21 (D) 20, u 3/ 10, 1/ () 5i 3j () 9i (C) 6j 6. () 6i 4j () 2i 6j (C) 2i 13j 7. u 4,900 lb, v 5,000 lb EXERCISE 7-4 In Problems 1 6, represent each geometric vector, with endpoints as indicated, as an algebraic vector in the form a, b. 1. (0, 0), (7, 2) 2. (5, 3), (0, 0) 3. (4, 0), (0, 8) 4. (0, 5), (6, 0) 5. (9, 4), (7, 5) 6. ( 6, 3), (9, 1) In Problems 7 12, find the magnitude of each vector , , , , , , 123 In Problems 13 16, find: () u v () u v (C) 2u v 3w 13. u 2, 1, v 1, 3, w 3, u 1, 2, v 3, 2, w 0, u 4, 1, v 2, 2, w 0, u 3, 2, v 2, 2, w 3, 0 In Problems 17 22, epress v in terms of the i and j unit vectors , , , , v, where (2, 3) and ( 3, 1) 22. v, where ( 2, 1) and (0, 2) In Problems 23 28, let u 3i 2j, v 2i 4j, and w 2i, and perform the indicated operations. 23. u v 24. u v 25. 2u 3v 26. 3u 2v 27. 2u v 2w 28. u 3v 2w In Problems 29 32, find a unit vector u with the same direction as v. 29. v 1, v 2, v 12, v 7, 24 In Problems 33 36, determine whether the statement is true or false. If true, eplain wh. If false, give a countereample. 33. If u is a scalar multiple of v, then u and v have the same direction. 34. If u and v are nonzero vectors that have the same direction, then u is a scalar multiple of v. 35. The sum of two unit vectors is a unit vector. 36. If u is a unit vector and k is a scalar, then the magnitude of ku is k. C In Problems 37 44, let u a, b, v c, d, and w e, f be vectors and m and n be scalars. Prove each of the following vector properties using appropriate properties of real numbers and the definitions of vector addition and scalar multiplication. 37. u (v w) (u v) w 38. u v v u 39. u 0 u 40. u ( u) (m n)u mu nu 42. m(u v) mu mv 43. m(nu) (mn)u 44. 1u u
10 540 7 dditional Topics in Trigonometr PPLICTINS In Problems 45 52, compute all answers to 3 significant digits. 45. Static Equilibrium. unicclist at a certain point on a tightrope deflects the rope as indicated in the figure. If the total weight of the cclist and the uniccle is 155 pounds, how much tension is in each part of the cable? 49. Static Equilibrium. 400-pound sign is suspended as shown in figure (a). The corresponding force diagram (b) is formed b observing the following: Member is pushing at and is under compression. This pushing force also can be thought of as the force vector a pulling to the right at. The force vector b reflects the fact that member C is under tension that is, it is pulling at. The force vector c corresponds to the weight of the sign pulling down at. Find the magnitudes of the forces in the rigid supporting members; that is, find a and b in the force diagram (b). C 2 ards b 155 pounds 1 ard a 46. Static Equilibrium. Repeat Problem 45 with the left angle 4.2, the right angle 5.3, and the total weight 112 pounds. 47. Static Equilibrium. weight of 1,000 pounds is suspended from two cables as shown in the figure. What is the tension in each cable? 400 pounds (a) (b) c Static Equilibrium. weight of 1,000 kilograms is supported as shown in the figure. What are the magnitudes of the forces on the members and C? 1,000 pounds 1 meter C 2 meters 48. Static Equilibrium. weight of 500 pounds is supported b two cables as illustrated. What is the tension in each cable? kilograms 500 pounds 51. Static Equilibrium. 1,250-pound weight is hanging from a hoist as indicated in the figure on the net page. What are the magnitudes of the forces on the members and C?
11 7-5 Polar Coordinates and Graphs 541 C 10.6 feet 52. Static Equilibrium. weight of 5,000 kilograms is supported as shown in the figure. What are the magnitudes of the forces on the members and C? 12.5 feet C 5 meters 1,250 pounds 6 m 5,000 kilograms Figure for 51 Figure for 52 SECTIN 7-5 Polar Coordinates and Graphs Polar Coordinate Sstem Converting from Polar to Rectangular Form, and Vice Versa Graphing Polar Equations Some Standard Polar Curves pplication Up until now we have used onl the rectangular coordinate sstem. ther coordinate sstems have particular advantages in certain situations. f the man that are possible, the polar coordinate sstem ranks second in importance to the rectangular coordinate sstem and forms the subject matter for this section. Pole Polar Coordinate Sstem Polar ais r P(r, ) FIGURE 1 Polar coordinate sstem. To form a polar coordinate sstem in a plane (see Fig. 1), start with a fied point and call it the pole, or origin. From this point draw a half line, or ra (usuall horizontal and to the right), and call this line the polar ais. If P is an arbitrar point in a plane, then associate polar coordinates (r, ) with it as follows: Starting with the polar ais as the initial side of an angle, rotate the terminal side until it, or the etension of it through the pole, passes through the point. The coordinate in (r, ) is this angle, in degree or radian measure. The angle is positive if the rotation is counterclockwise and negative if the rotation is clockwise. The r coordinate in (r, ) is the directed distance from the pole to the point P. It is positive if measured from the pole along the terminal side of and negative if measured along the terminal side etended through the pole. Figure 2 illustrates a point P with three different sets of polar coordinates. Stud this figure carefull. The pole has polar coordinates (0, ) for arbitrar. For eample, (0, 0 ), (0, /3), and (0, 371 ) are all coordinates of the pole. FIGURE 2 Polar coordinates of a point. 5 P 4 4, P ( 4, 225 ) 5 P 3 4, (a) (b) (c)