1. VECTORS AND THE GEOMETRY OF SPACE 12

2. 12.2 Vectors VECTORS AND THE GEOMETRY OF SPACE In this section, we will learn about: Vectors and their applications.

3. The term vector is used by scientists to indicate a quantity (such as displacement or velocity or force) that has both magnitude and direction. VECTOR

4. A vector is often represented by an arrow or a directed line segment.  The length of the arrow represents the magnitude of the vector.  The arrow points in the direction of the vector. REPRESENTING A VECTOR

5. We denote a vector by either:  Printing a letter in boldface (v)  Putting an arrow above the letter ( ) DENOTING A VECTOR

6. For instance, suppose a particle moves along a line segment from point A to point B. VECTORS

7. The corresponding displacement vector v has initial point A (the tail) and terminal point B (the tip).  We indicate this by writing v =. VECTORS

8. Notice that the vector u = has the same length and the same direction as v even though it is in a different position.  We say u and v are equivalent (or equal) and write u = v. VECTORS

9. The zero vector, denoted by 0, has length 0.  It is the only vector with no specific direction. ZERO VECTOR

10. Suppose a particle moves from A to B. So, its displacement vector is. COMBINING VECTORS

11. Then, the particle changes direction, and moves from B to C—with displacement vector. COMBINING VECTORS

12. The combined effect of these displacements is that the particle has moved from A to C. COMBINING VECTORS

13. The resulting displacement vector is called the sum of and. We write: COMBINING VECTORS

14. In general, if we start with vectors u and v, we first move v so that its tail coincides with the tip of u and define the sum of u and v as follows. ADDING VECTORS

15. If u and v are vectors positioned so the initial point of v is at the terminal point of u, then the sum u + v is the vector from the initial point of u to the terminal point of v. VECTOR ADDITION—DEFINITION

16. The definition of vector addition is illustrated here. VECTOR ADDITION

17. You can see why this definition is sometimes called the Triangle Law. TRIANGLE LAW

18. Here, we start with the same vectors u and v as earlier and draw another copy of v with the same initial point as u. VECTOR ADDITION

19. Completing the parallelogram, we see that: u + v = v + u VECTOR ADDITION

20. This also gives another way to construct the sum:  If we place u and v so they start at the same point, then u + v lies along the diagonal of the parallelogram with u and v as sides.

21. PARALLELOGRAM LAW This is called the Parallelogram Law.

22. Draw the sum of the vectors a and b shown here. VECTOR ADDITION Example 1

23. VECTOR ADDITION First, we translate b and place its tail at the tip of a—being careful to draw a copy of b that has the same length and direction. Example 1

24. Then, we draw the vector a + b starting at the initial point of a and ending at the terminal point of the copy of b. VECTOR ADDITION Example 1

25. VECTOR ADDITION Alternatively, we could place b so it starts where a starts and construct a + b by the Parallelogram Law. Example 1

26. It is possible to multiply a vector by a real number c. MULTIPLYING VECTORS

27. In this context, we call the real number c a scalar—to distinguish it from a vector. SCALAR

28. For instance, we want 2v to be the same vector as v + v, which has the same direction as v but is twice as long. In general, we multiply a vector by a scalar as follows. MULTIPLYING SCALARS

29. If c is a scalar and v is a vector, the scalar multiple cv is: The vector whose length is |c| times the length of v and whose direction is the same as v if c > 0 and is opposite to v if c < 0.  If c = 0 or v = 0, then cv = 0. SCALAR MULTIPLICATION—DEFINITION

30. The definition is illustrated here.  We see that real numbers work like scaling factors here.  That’s why we call them scalars. SCALAR MULTIPLICATION

31. Notice that two nonzero vectors are parallel if they are scalar multiples of one another. SCALAR MULTIPLICATION

32. In particular, the vector –v = (–1)v has the same length as v but points in the opposite direction.  We call it the negative of v. SCALAR MULTIPLICATION

33. SUBTRACTING VECTORS By the difference u – v of two vectors, we mean: u – v = u + (–v)

34. So, we can construct u – v by first drawing the negative of v, –v, and then adding it to u by the Parallelogram Law. SUBTRACTING VECTORS

35. Alternatively, since v + (u – v) = u, the vector u – v, when added to v, gives u. SUBTRACTING VECTORS

36. So, we could construct u by means of the Triangle Law. SUBTRACTING VECTORS

37. If a and b are the vectors shown here, draw a – 2b. SUBTRACTING VECTORS Example 2

38. First, we draw the vector –2b pointing in the direction opposite to b and twice as long. Next, we place it with its tail at the tip of a. SUBTRACTING VECTORS Example 2

39. Finally, we use the Triangle Law to draw a + (–2b). SUBTRACTING VECTORS Example 2

40. COMPONENTS For some purposes, it’s best to introduce a coordinate system and treat vectors algebraically.

41. COMPONENTS Let’s place the initial point of a vector a at the origin of a rectangular coordinate system.

42. Then, the terminal point of a has coordinates of the form (a 1, a 2 ) or (a 1, a 2, a 3 ).  This depends on whether our coordinate system is two- or three-dimensional. COMPONENTS

43. These coordinates are called the components of a and we write: a = ‹a 1, a 2 › or a = ‹a 1, a 2, a 3 ›

44. We use the notation ‹ a 1, a 2 › for the ordered pair that refers to a vector so as not to confuse it with the ordered pair (a 1, a 2 ) that refers to a point in the plane. COMPONENTS

45. For instance, the vectors shown here are all equivalent to the vector whose terminal point is P(3, 2). COMPONENTS

46. What they have in common is that the terminal point is reached from the initial point by a displacement of three units to the right and two upward. COMPONENTS

47. We can think of all these geometric vectors as representations of the algebraic vector a = ‹3, 2›. COMPONENTS

48. POSITION VECTOR The particular representation from the origin to the point P(3, 2) is called the position vector of the point P.

49. POSITION VECTOR In three dimensions, the vector a = = ‹ a 1, a 2, a 3 › is the position vector of the point P(a 1, a 2, a 3 ).

50. COMPONENTS Let’s consider any other representation of a, where the initial point is A(x 1, y 1, z 1 ) and the terminal point is B(x 2, y 2, z 2 ).

51. COMPONENTS Then, we must have: x 1 + a 1 = x 2, y 1 + a 2 = y 2, z 1 + a 3 = z 2 Thus, a 1 = x 2 – x 1, a 2 = y 2 – y 1, a 3 = z 2 – z 1  Thus, we have the following result.

52. COMPONENTS Given the points A(x 1, y 1, z 1 ) and B(x 2, y 2, z 2 ), the vector a with representation is: a = ‹ x 2 – x 1, y 2 – y 1, z 2 – z 1 › Equation 1

53. COMPONENTS Find the vector represented by the directed line segment with initial point A(2, –3, 4) and terminal point B(–2, 1, 1).  By Equation 1, the vector corresponding to is: a = ‹ –2 –2, 1 – (–3), 1 – 4 › = ‹ –4, 4, –3 › Example 3

54. LENGTH OF VECTOR The magnitude or length of the vector v is the length of any of its representations.  It is denoted by the symbol |v| or║v║.

55. LENGTH OF VECTOR By using the distance formula to compute the length of a segment, we obtain the following formulas.

56. LENGTH OF 2-D VECTOR The length of the two-dimensional (2-D) vector a = ‹a 1, a 2 › is:

57. LENGTH OF 3-D VECTOR The length of the three-dimensional (3-D) vector a = ‹ a 1, a 2, a 3 › is:

58. ALGEBRAIC VECTORS How do we add vectors algebraically?

59. ADDING ALGEBRAIC VECTORS The figure shows that, if a = ‹ a 1, a 2 › and b = ‹ b 1, b 2 ›, then the sum is a + b = ‹ a 1 + b 1, a 2 + b 2 › at least for the case where the components are positive.

60. ADDING ALGEBRAIC VECTORS In other words, to add algebraic vectors, we add their components.

61. SUBTRACTING ALGEBRAIC VECTORS Similarly, to subtract vectors, we subtract components.

62. ALGEBRAIC VECTORS From the similar triangles in the figure, we see that the components of ca are ca 1 and ca 2.

63. MULTIPLYING ALGEBRAIC VECTORS So, to multiply a vector by a scalar, we multiply each component by that scalar.

64. 2-D ALGEBRAIC VECTORS If a = ‹ a 1, a 2 › and b = ‹ b 1, b 2 ›, then a + b = ‹ a 1 + b 1, a 2 + b 2 › a – b = ‹ a 1 – b 1, a 2 – b 2 › ca = ‹ ca 1, ca 2 ›

65. 3-D ALGEBRAIC VECTORS Similarly, for 3-D vectors,

66. ALGEBRAIC VECTORS If a = ‹4, 0, 3› and b = ‹–2, 1, 5›, find: |a| and the vectors a + b, a – b, 3b, 2a + 5b Example 4

67. ALGEBRAIC VECTORS Example 4 |a||a|

68. ALGEBRAIC VECTORS Example 4

69. ALGEBRAIC VECTORS Example 4

70. ALGEBRAIC VECTORS Example 4

71. ALGEBRAIC VECTORS Example 4

72. COMPONENTS We denote:  V 2 as the set of all 2-D vectors  V 3 as the set of all 3-D vectors

73. COMPONENTS More generally, we will later need to consider the set V n of all n-dimensional vectors.  An n-dimensional vector is an ordered n-tuple a = ‹a 1, a 2, …, a n › where a 1, a 2, …, a n are real numbers that are called the components of a.

74. COMPONENTS Addition and scalar multiplication are defined in terms of components just as for the cases n = 2 and n = 3.

75. PROPERTIES OF VECTORS If a, b, and c are vectors in V n and c and d are scalars, then

76. PROPERTIES OF VECTORS These eight properties of vectors can be readily verified either geometrically or algebraically.

77. PROPERTY 1 For instance, Property 1 can be seen from this earlier figure.  It’s equivalent to the Parallelogram Law.

78. PROPERTY 1 It can also be seen as follows for the case n = 2: a + b = ‹a 1, a 2 › + ‹b 1, b 2 › = ‹a 1 + b 1, a 2 + b 2 › = ‹b 1 + a 1, b 2 + a 2 › = ‹ b 1, b 2 › + ‹ a 1, a 2 › = b + a

79. PROPERTY 2 We can see why Property 2 (the associative law) is true by looking at this figure and applying the Triangle Law several times, as follows.

80. PROPERTY 2 The vector is obtained either by first constructing a + b and then adding c or by adding a to the vector b + c.

81. VECTORS IN V 3 Three vectors in V 3 play a special role.

82. VECTORS IN V 3 Let i = ‹1, 0, 0› j = ‹0, 1, 0› k = ‹0, 0, 1›

83. STANDARD BASIS VECTORS These vectors i, j, and k are called the standard basis vectors.  They have length 1 and point in the directions of the positive x-, y-, and z-axes.

84. STANDARD BASIS VECTORS Similarly, in two dimensions, we define: i = ‹ 1, 0 › j = ‹ 0, 1 ›

85. STANDARD BASIS VECTORS If a = ‹a 1, a 2, a 3 ›, then we can write:

86. STANDARD BASIS VECTORS Equation 2

87. STANDARD BASIS VECTORS Thus, any vector in V 3 can be expressed in terms of i, j, and k.

88. STANDARD BASIS VECTORS For instance, ‹1, –2, 6 › = i – 2j + 6k

89. STANDARD BASIS VECTORS Similarly, in two dimensions, we can write: a = ‹a 1, a 2 › = a 1 i + a 2 j Equation 3

90. COMPONENTS Compare the geometric interpretation of Equations 2 and 3 with the earlier figures.

91. COMPONENTS If a = i + 2j – 3k and b = 4i + 7k, express the vector 2a + 3b in terms of i, j, and k. Example 5

92. COMPONENTS Using Properties 1, 2, 5, 6, and 7 of vectors, we have: 2a + 3b = 2(i + 2j – 3k) + 3(4i + 7k) = 2i + 4j – 6k + 12i + 21k = 14i + 4j + 15k Example 5

93. UNIT VECTOR A unit vector is a vector whose length is 1.  For instance, i, j, and k are all unit vectors.

94. UNIT VECTORS In general, if a ≠ 0, then the unit vector that has the same direction as a is: Equation 4

95. UNIT VECTORS In order to verify this, we let c = 1/|a|.  Then, u = ca and c is a positive scalar; so, u has the same direction as a.  Also,

96. UNIT VECTORS Find the unit vector in the direction of the vector 2i – j – 2k. Example 6

97. UNIT VECTORS The given vector has length  So, by Equation 4, the unit vector with the same direction is: Example 6

98. APPLICATIONS Vectors are useful in many aspects of physics and engineering.  In Chapter 13, we will see how they describe the velocity and acceleration of objects moving in space.  Here, we look at forces.

99. FORCE A force is represented by a vector because it has both a magnitude (measured in pounds or newtons) and a direction.  If several forces are acting on an object, the resultant force experienced by the object is the vector sum of these forces.

100. FORCE A 100-lb weight hangs from two wires. Find the tensions (forces) T 1 and T 2 in both wires and their magnitudes Example 7

101. FORCE First, we express T 1 and T 2 in terms of their horizontal and vertical components. Example 7

102. FORCE From the figure, we see that: T 1 = –|T 1 | cos 50° i + |T 1 | sin 50° j T 2 = |T 2 | cos 32° i + |T 2 | sin 32° j E. g. 7—Eqns. 5 & 6

103. FORCE The resultant T 1 + T 2 of the tensions counterbalances the weight w. So, we must have: T 1 + T 2 = –w = 100 j Example 7

104. FORCE Thus, (–|T 1 | cos 50° + |T 2 | cos 32°) i + (|T 1 | sin 50° + |T 2 | sin 32°) j = 100 j Example 7

105. FORCE Equating components, we get: –|T 1 | cos 50° + |T 2 | cos 32° = 0 |T 1 | sin 50° + |T 2 | sin 32° = 100 Example 7

106. FORCE Solving the first of these equations for |T 2 | and substituting into the second, we get: Example 7

107. FORCE So, the magnitudes of the tensions are: Example 7

108. FORCE Substituting these values in Equations 5 and 6, we obtain the tension vectors T 1 ≈ –55.05 i + 65.60 j T 2 ≈ 55.05 i + 34.40 j Example 7