8.0 Definition and the concept of a vector:

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1 Chapter 8: Vectors In this chapter, we will study: 80 Definition and the concept of a ector 81 Representation of ectors in two dimensions (2D) 82 Representation of ectors in three dimensions (3D) 83 Operations with ectors (calculations): 83A Multiplication of a ector with a scalar 83B Multiplication of a ector with (-1) 83C Addition of ectors 83D Subtraction of ectors 83E Unit ectors 83F The angle between two ectors 80 Definition and the concept of a ector: The quantities studied so far in this class are scalar quantities Scalar quantities are characterized only by their size In this chapter we study ectors Vectors are characterized not only by size, but also by direction Definition 1: A ector is a directed line segment, as shown in Figure 1 below Figure 1: A ector A ector has therefore a magnitude (size), and a direction (the line on which this ector sits and its arrowhead) Note that a ector does not hae a fixed position (that is, ectors which are parallel to one another but are equal in magnitude and direction are considered identical) Practically, ectors are frequently used in mechanics: for example the elocity of an object and a force which acts on an object are ector quantities In contrast, the mass (of an object) and its temperature are scalar quantities 1

2 Note: In this chapter we represent a ector either through a bolded case letter (for example as u ), or through the notation: AB ( AB represents the ector which starts at A and ends at B) A scalar quantity is denoted by an un-bolded letter (such as a ) 81 Representation of ectors in two dimensions (2D) In two dimensions, a ector in components form is ab, For example: 4,2 4i2j, which is represented in Figure 2 below: which can be written asa, b a* ib* j Figure 2: The ector 4,2 4i2j If we place the ector (4,2) at origin (its tail is at origin) then 4,2 OP with P 4,2 For a generic 2D ector ab,, we define its magnitude (size) as the scalar quantity gien by: (1) and it represents the length of the segment determined by this ector (as we can deduce from Figure 2 using Pythagoras theorem) 2 2 For example: 4, Aside from the coordinates form used aboe, a ector ab, in polar form), as shown in Figure 3 below: a b 2 2 can also be represented in magnitude-direction form (or Figure 2: The ector 4,2 in magnitude direction form In magnitude-direction form:,, the ector ab, is characterized by its magnitude: a b 2 2 and by its direction (the angle which the ector makes with the positie Ox axis) 2

3 We want now to find the magnitude-direction form (the angle ) of a ector gien in component form By the definition of the trigonometric functions sin and cos (see sections 71 and 73), we hae that: (2) sin cos b a To find, using formulas (2), we deduce: 1 b (3) tan k, where: a k 0, when a, b Q I k 1, when a, b Q II or Q III k 2, when a, b Q IV Therefore, to find the magnitude-direction form of a ector gien in component form, use equations (1) and (3) Conersely, if the polar form of a ector is gien, (2), written as:, to find its coordinate form ( a and b), use relationships (4) a b cos sin Example: Do Examples 81, 82, 83 and 84 from pages in the textbook 82 Representation of ectors in three dimensions (3D) In 3 dimensions, a 3 coordinates point is represented as in Figure 3 below: Figure 3: Graphical representation of the point (3,4,1) in 3D 3

4 A generic 3D ector abc,, is gien by: (4) a, b, c a* i b* j c* k The size of the 3D ector abc,, is gien by: (5) a b c Note that two ectors and w are equal ( = w ) if: = w ; the direction of is equal to the direction of w; If ectors are gien in component form, this means that each component of is equal to each corresponding component of w As mentioned in Definition 1 in 80, a ector does not hae a fixed position Howeer, if the origin of a ector (its tail) is fixed, then the position of that ector is fixed as well A ector which starts at origin O (0,0,0) is called the position ector For example, the position ector which ends at point P (3,4,1) is: OP 3i 4jk (shown aboe in Figure 3) In general: (5) AB OB OA, or in a different notation: (6) OB OA AB (this formula is easy to memorize: if we consider that the head of the first ector collapses with the tail of the second ector to create the ector OB ) Exercises: Do Examples 85 and 86 from pages 260 and 261 in the textbook and from Exercise set 8A do problems 1, 2 i, iii, 3 ii, i, i, 4 (choose 3 sub-points) and 5 83A Multiplication of a ector with a scalar: Definition 2: 83 Operations with ectors (calculations with ectors) For a gien scalar 0 and a gien ector, the ector is defined as the ector which has the same direction with, but a magnitude of For example, the ector 2a has the same direction with a and a magnitude of 2 a, as shown in Figure 4 below: 4

5 Figure 4: Vectors a and 2a When written in component form, when a, b, c then a, b, c a, b, c For example: 2 3, 5,1 6, 10, 2 or : 2 3i 5j k 6i 10j 2k 83B Multiplication of a ector with (-1) (The negatie of a ector): Definition 3: For a gien ector, the ector is the ector which has the same magnitude with and a direction opposite to This is shown in Figure 5 below: Figure 5: Vectors a and a In component form, when multiplying a by (-1) each component of a is multiplied by (-1) For example: 3, 5,1 3,5, 1, or 1,1 1, 1 (represent these ectors on the Oxyz and Oxy axes respectiely to isualize) 83C Addition of ectors: Definition 4: Algebraically, we define: (7) u, u, u,, u, u, u Example: 2, 3 3,5 5, 2 u +, and similarly in 2D Geometrically, the sum of two ectors represents the resultant of the displacement, when the tail of one ector is placed at the head of the other (this can be thought as the corresponding diagonal of the parallelogram formed with the two ectors), 5

6 as shown in Figure 6 below: Figure 6: The geometrical representation of 2, 3 3,5 (5, 2) Note: when adding ectors, it is essential to place the tail of one ector at the head of the ector u, and then obtain the sum u + as the ector formed from the tail ector u to the head of ector This conention should be followed when adding more than two ectors, as in Figure 7 below: Figure 7: resultant = leg 1+leg 2+leg 3 This conention can also be understood algebraically when using the notation: AC AB BC 83D Subtraction of ectors Definition 5: Define (8) u - = u +(-), and use the meaning of - as in 83B and of u +(-) as in 83C Geometrically, the difference u- represents the other diagonal (than the one gien by the sum u + ) of the parallelogram formed by the ectors u and Example: Do Example 88 from page 264 of the textbook, Actiity 81 and Example 89 on page 266 6

7 83E Unit ectors Definition 6: For a gien ector a, we define the unit ector in the direction of a as: a U= a Note that the ector U has a magnitude of 1 (it is a unit ector) Example: 1 To find the unit ector in the direction of a = 3i + 5j, calculate Do Example 810 (page 268) of the textbook a 1 U= a 34 a 3,5 Exercises: From problem set 8B (page 269) of the textbook do problems 1 (choose 3 sub-points), 2 (choose 3 sub-points), 3,4,5, 8 and 10 83F The angle between two ectors Consider two ectors (here in 2D) a and b as shown in Figure 8 below Figure 8: Vectors a and band the angle between a and b It can be shown using the law of cosines in triangle OAB (see the proof on page 271 of the textbook) that: (9) cos ab a b a b a b a b Similarly, in 3D: (10) cos ab a b a b a b a b a b The expression ab a b a b (in 2D) and (12) ab a1b 1 a2b2 +a3b 3 (in 2D) (11) is called the scalar product of ectors a and b 7

8 1 Therefore: (11) cos ab ab Note: Two ectors (11)-(12) aboe a and bare perpendicular a b if cos 0 ab 0, where abis gien by the formulas Example: Do Examples 811, 812 and 813 from pages in the textbook Exercise: From Exercise set 8C do problems 1 (choose 3 sub-points)2, 5 and 6 to 12 As a summary of this chapter, read the following Key Points: 8

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