1. Introduction to Vectors Section 12.1 1

2. Vectors are an essential tool in physics and a very significant part of mathematics. Their primary application was to represent forces and the operation called “vector addition” corresponds to the combining of various forces. We can represent physical quantities like temperature, distance, area and speed by a single number indicating magnitude called “scalar quantities”. Other physical quantities posses the properties of magnitude and direction. We define force needed to pull a truck up a 10° slope by its magnitude and direction. Force, displacement, velocity, acceleration, etc are quantities that cannot be described by a single number. These are called vector quantities. 2 EX: If you wish to fly to from London to Paris and I tell you that the distance is 340 km, this piece of information is useless until I tell you what direction you need to travel.

3. Speed and velocity have different meanings. Speed is a scalar quantity that refers to “how fast an object is moving”. Velocity is a vector quantity that refers to “the rate at which an object changes its position. So you might describe it as 55 km/hr east. Thus an airplane moving west with a speed of 600 km/hr has a velocity of 600 km/hr west. 3

4. Vector is a quantity that has both magnitude (length) and direction. It is represented geometrically by a directed line segment. A directed line segment with an initial point P and a terminal point at Q a When we handwrite vectors, we cannot use boldface, so the convention is to use the arrow notation. 4

5. The length of the line segment is the magnitude. If the vector has its initial point at the origin, it is in standard position. The direction is the directed angle between the positive x-axis and the vector. 5

6. 6 In a column vector, the x represents a movement in the positive x direction and the y represents a movement in the positive y direction. Component form: A vector in standard position can be uniquely represented by the coordinates of its terminal point (u 1, u 2 ). This can also be written as

7. Unit Vectors – another way to represent vectors The unit vectors (1, 0) and (0, 1) are called the standard unit vectors and are denoted by i = (1, 0) and j = (0, 1) Therefore the vector 3i + 4j means a movement of 3 units in the positive x axis and 4 in the positive y directions 7

8. 8 The unit vector in the direction of x-axis is i, in the y-axis is j and in the direction of z is k. The vectors i, j, k are called base vectors. In two dimensions i =In two dimensions j = In three dimensions i = In three dimensions j = Therefore, k in three dimensions is k =

9. 9 EX: Write a = in unit vector form. ANS: a = 6i – 7j EX: Write –i + 5k in column vector form. ANS:

10. 10

11. Example 5 – Writing a Linear Combination of Unit Vectors Let u be the vector with initial point (2, –5) and terminal point (–1, 3). Write u as a linear combination of the standard unit vectors i and j. Solution: Begin by writing the component form of the vector u. 11

12. Example 5 – Solution This result is shown graphically in Figure 6.28. cont’d Figure 6.28 12

13. TRY: Write the ordered pair as a vector and the find the magnitude from C(7,-3) to D(-2,-1). (7, -7) 13

14. This formula is also used to find the distance between two points in space. 14

15. 15

16. Two vectors are equivalent if and only if they have the same magnitude and direction. 16

17. 17 Two vectors are Parallel if one is a scalar multiple of the other. So and are parallel if, where k is a scalar quantity. This can also be written as a = kb EX: For what values of t and s are these two vectors parallel? m = 3i + tj – 6k and n = 9i – 12j + sk t = -4 and s = -18

18. HW: P 410 12A 1ac, 2, 3, 4ac, 5ac and P 413 12B 1, 2, 4, 5 18 Also: P 166 6C 2 and P 168 6D 1b

19. The two basic vector operations are scalar multiplication and vector addition. Geometrically, the product of a vector v and a scalar k is the vector that is | k | times as long as v. If k is positive, then k v has the same direction as v, and if k is negative, then k v has the opposite direction of v, 19

20. 20

21. The sum of two or more vectors is called the resultant. It can be found using either the parallelogram method or the triangle method. 21

22. 22

23. Two vectors are opposite if they have the same magnitude and opposite directions. (can be used for subtraction) 23 Show subtraction on parallelogram w - v

24. HW: Worksheet 12.1 and P 416 12C 1-5 and P 417 12D 1, 2 (example #7 is #1) 24 Also: P 169 6E 1, 4, 6

25. Unit Vectors In many applications of vectors, it is useful to find a unit vector that has the same direction as a given nonzero vector v. To do this, you can divide v by its length to obtain Note that u is a scalar multiple of v. The vector u has a magnitude of 1 and the same direction as v. The vector u is called a unit vector in the direction of v. Unit vector in direction of v 25

26. Consider the vector u= 3i + 4j To find the magnitude of u, |u|, we use the distance formula. Therefore, |u| = 5. If we divide the vector u by its magnitude we get another vector that is parallel to u, since they are scalar multiples of each other. This new vector is This vector is called the unit vector because its magnitude is equal to 1 Therefore to find a unit vector in the same direction as a given vector, just divide that vector by its own magnitude 26

27. Example – Finding a Unit Vector Find a unit vector in the direction of and verify that the result has a magnitude of 1. Solution: The unit vector in the direction of v is 27

28. Vector Operations 28

29. Example 3 – Vector Operations Let v = (-2, 5) and w = (3, 4) and find each of the following vectors. a. 2vb. w – vc. v + 2w Solution: a. Because you have A sketch of 2v is shown in Figure 6.24. Figure 6.24 29

30. Example 3(b) – Solution The difference of w and v is A sketch of w – v is shown in Figure 6.25. Note that the figure shows the vector difference w – v as the sum w + (– v). Figure 6.25 cont’d 30

31. Example 3(c) – Solution The sum of v and 2w is A sketch of v + 2w is shown in Figure 6.26. Figure 6.26 cont’d 31

32. 32

33. 33 The zero vector Two Dimensional 0 = Three Dimensional 0 = Given Triangle PQR, then must be equal to zero as the overall journey results in a return to the starting point. P Q R Equilibrium is the name for the state where a number of forces are in balance – their resultant is zero.

34. 34 HW: P 418 12E 1-4 P 420 12F 2, 4-8 P 422 12G 1 d-f, 2ce, 3c, 4, 6 BRING BOOKS!!!!! Also: P 171 6F 1 (do 2 of them), 2a, 3a

35. 35

36. 36

37. 37

38. Scalar Product/Dot Product of two vectors a · b = |a||b|cosθ Note that the result of a dot product is a scalar not a vector. If a and b are perpendicular then the dot product is equal to 0. If a and b are parallel then the dot product is equal to ±|a||b| If vectors are in component form then the dot product is 38

39. The Angle between two vectors Since a · b = |a||b|cosθ, then you can see that Example: Find the angle between the following two vectors: v = -3i + 3j and 2i – 4j Answer: θ = 161.57 ⁰ 39

40. 40 Two vectors are perpendicular/orthoganal if a  b = 0 Two vectors are parallel if a  b = |a||b| or a multiple of a  b Two vectors are coincident/same vector if a  b = a 2

41. 41 HW: P 424 12H 1-3 and P 428 12I 1 c-e, 2 c-e, 3-15 (no calc on #6) Show typos on Pg 425 Also: P 173 6G 3, 5

42. Section 12.4 Vector Equation of a Line 42

43. 43

44. 44

45. 45

46. 46

47. 47

48. The angle between the lines l and m is the angle between these two direction vectors 48

49. Application to Motion Problems r = a + tb where t represents time a represents the initial position b represent velocity |b| represents speed 49

50. 50 Show that the lines intersect and find the coordinates of the point of intersection. 3 + s = 6 s =2 + 4t -1 + s = 8t S = 3 and t = ¼ is consistent on all

51. 51 HW: P 432 12J 1-4 a & c only, 7 a,b,d 8a, 9-10 and P 435 12K21-6, 8