1. © 2013 Pearson Education, Inc. CHAPTER 12. B.1 Geometric Operations with Vectors

2. © 2013 Pearson Education, Inc.

3. Remember…. When drawing vectors… length = magnitude (with scale) angle = direction of the vector quantity. When drawing and moving vectors, these two characteristics must be maintained

4. © 2013 Pearson Education, Inc.  Vector Addition = finding the sum or Resultant of two or more vector quantities

5. © 2013 Pearson Education, Inc. Resultant Vector  A Resultant vector is the vector that results from the addition of two or more vectors VAVA VBVB VRVR

6. © 2013 Pearson Education, Inc. Tip to Tail Method in-line examples Place the tail of one vector at the tip of the other. The vector sum (also called the resultant) is shown in red. It starts where the black vector began and goes to the tip of the blue one. In these cases, the vector sum represents the net force. You can only add or subtract magnitudes when the vectors are in-line! 16 m/s 20 m/s 4 m/s 20 m/s 16 m/s 12 m/s 9 m/s 12 m/s 21 m/s

7. © 2013 Pearson Education, Inc. Tip to Tail – 2 Vectors 5 m 2 m To add the red and blue displacement vectors first note: Vectors can only be added if they are of the same quantity—in this case, displacement. The magnitude of the resultant must be less than 7 m (5 + 2 = 7) and greater than 3 m (5 - 2 = 3). 5 m 2 m blue + black Interpretation: Walking 5 m in the direction of the blue vector and then 2 m in the direction of the black one is equivalent to walking in the direction of the red vector. The distance walked this way is the red vector’s magnitude.

8. © 2013 Pearson Education, Inc. Commutative Property blue + black black + blue As with scalars (ordinary numbers), the order of addition is irrelevant with vectors. Note that the resultant (black vector) is the same magnitude and direction in each case. If you’ve drawn everything to scale, and drawn the angles correctly, then you can simply measure the resultant vector and (using your scale) determine its magnitude.

9. © 2013 Pearson Education, Inc. Tip to Tail – 3 Vectors We can add 3 or more vectors by placing them tip to tail in any order, so long as they are of the same type (force, velocity, displacement, etc.). blue + green + black

10. © 2013 Pearson Education, Inc. The Zero Vector zero vector written 0

11. © 2013 Pearson Education, Inc. Page 281 1a. q p + q

12. © 2013 Pearson Education, Inc. Page 281 1f. q p + q

13. © 2013 Pearson Education, Inc. HOMEWORK page 281(12B.1) Numbers 1 – 5

14. © 2013 Pearson Education, Inc. CHAPTER 12. B.2

15. Subtracting Vectors A – B = ? Chapter 12. B.2

16. © 2013 Pearson Education, Inc. Example Problem What is 30 m/s north minus 15 m/s east?

17. © 2013 Pearson Education, Inc. Example Problem What is 30 m/s north minus 15 m/s east? Question: How many vectors are there in this problem?

18. © 2013 Pearson Education, Inc. Example Problem What is 30 m/s north minus 15 m/s east? Question: How many vectors are there in this problem?

19. © 2013 Pearson Education, Inc. Words  Math What is 30 m/s north minus 15 m/s east?

20. © 2013 Pearson Education, Inc. What is 30 m/s north minus 15 m/s east? Mathematically A – B = ? Words  Math

21. © 2013 Pearson Education, Inc. What is 30 m/s north minus 15 m/s east? Mathematically A – B = ? Words  Math The Problem: We only know how to add vectors.

22. © 2013 Pearson Education, Inc. The Trick: Make it Addition A – B = A + (– B)

23. © 2013 Pearson Education, Inc. The Trick: Make it Addition A – B = A + (– B) Guess: what is – B?

24. © 2013 Pearson Education, Inc. Time Out! C This is C

25. © 2013 Pearson Education, Inc. Time Out! C Point in the direction of – C!

26. © 2013 Pearson Education, Inc. Time Out! C Point in the direction of – C!

27. © 2013 Pearson Education, Inc. Time Out! C Point in the direction of – C! - C

28. © 2013 Pearson Education, Inc. Continue

29. © 2013 Pearson Education, Inc. The Trick: Make it Addition A – B = A + (– B) - B is B in the opposite direction

30. © 2013 Pearson Education, Inc. The Trick: Make it Addition A – B = A + (– B) - B is B in the opposite direction

31. © 2013 Pearson Education, Inc. Graphical Addition of Vectors 1.Make a scale drawing. 2.Move the vectors tip- to-tail. 3.Draw the resultant vector (free tail to free tip). 4.Measure the length of the resultant with a ruler. 5.Measure the direction of the resultant with a protractor. A + (– B) 1 cm = 5 m/s

32. © 2013 Pearson Education, Inc. Graphical Addition of Vectors 1.Make a scale drawing. 2.Move the vectors tip- to-tail. 3.Draw the resultant vector (free tail to free tip). 4.Measure the length of the resultant with a ruler. 5.Measure the direction of the resultant with a protractor. A + (– B) 1) Which are tip-to-tail? 3) 2) 4)

33. © 2013 Pearson Education, Inc. Graphical Addition of Vectors 1.Make a scale drawing. 2.Move the vectors tip- to-tail. 3.Draw the resultant vector (free tail to free tip). 4.Measure the length of the resultant with a ruler. 5.Measure the direction of the resultant with a protractor. A + (– B) 1) Which are tip-to-tail? 3) 2) 4)

34. © 2013 Pearson Education, Inc. Graphical Addition of Vectors 1.Make a scale drawing. 2.Move the vectors tip- to-tail. 3.Draw the resultant vector (free tail to free tip). 4.Measure the length of the resultant with a ruler. 5.Measure the direction of the resultant with a protractor. A + (– B) 1) Draw the resultant for 1): from point ___ to point ___? 2) c b a

35. © 2013 Pearson Education, Inc. Graphical Addition of Vectors 1.Make a scale drawing. 2.Move the vectors tip- to-tail. 3.Draw the resultant vector (free tail to free tip). 4.Measure the length of the resultant with a ruler. 5.Measure the direction of the resultant with a protractor. A + (– B) 1) Draw the resultant for 1): from point ___ to point ___? 2) c b a ac

36. © 2013 Pearson Education, Inc. Graphical Addition of Vectors 1.Make a scale drawing. 2.Move the vectors tip- to-tail. 3.Draw the resultant vector (free tail to free tip). 4.Measure the length of the resultant with a ruler. 5.Measure the direction of the resultant with a protractor. A + (– B) 1) Draw the resultant for 2): from point ___ to point ___? 2) f d e

37. © 2013 Pearson Education, Inc. Graphical Addition of Vectors 1.Make a scale drawing. 2.Move the vectors tip- to-tail. 3.Draw the resultant vector (free tail to free tip). 4.Measure the length of the resultant with a ruler. 5.Measure the direction of the resultant with a protractor. A + (– B) 1) Draw the resultant for 2): from point ___ to point ___? 2) f d e fd

38. © 2013 Pearson Education, Inc. Graphical Addition of Vectors 1.Make a scale drawing. 2.Move the vectors tip- to-tail. 3.Draw the resultant vector (free tail to free tip). 4.Measure the length of the resultant with a ruler. 5.Measure the direction of the resultant with a protractor. A + (– B) 1) Notice: the resultant is the same for option 1) and option 2). 2)

39. © 2013 Pearson Education, Inc. Subtraction b a

40. © 2013 Pearson Education, Inc. Subtraction b a -b

41. © 2013 Pearson Education, Inc. Subtraction b a -b a+(-b)

42. © 2013 Pearson Education, Inc. Subtraction b a -b a+(-b)

43. © 2013 Pearson Education, Inc. Subtraction b a -b a+(-b) If a and b share the same initial point, the vector a-b is the vector from the terminal point of b to the terminal point of a

44. © 2013 Pearson Education, Inc. Geometrically, what is the difference? Let u and v be the vectors below. What is u – v? vectors u & v are in standard position Now, create the vector -v

45. © 2013 Pearson Education, Inc. Geometrically, what is the difference? Let u and v be the vectors below. What is u – v? Now, create the vector -v

46. © 2013 Pearson Education, Inc. draw the sum of u - v Geometrically, what is the difference? Let u and v be the vectors below. What is u – v?

47. © 2013 Pearson Education, Inc. draw the sum of u - v Geometrically, what is the difference? Let u and v be the vectors below. What is u – v?

48. © 2013 Pearson Education, Inc. HOMEWORK page 282(12B.2) Numbers 1 – 3 page 283(12B.3) Numbers 1 – 68(ALL)

49. Scalar Multiplication Chapter 12. B.4

50. © 2013 Pearson Education, Inc. The negative of a vector is just a vector going the opposite way. A number multiplied in front of a vector is called a scalar. It means to take the vector and add together that many times.

51. © 2013 Pearson Education, Inc. Using the vectors shown, find the following:

52. © 2013 Pearson Education, Inc. HOMEWORK page 285(12B.4) Numbers 1 – 62 (All) Skip #3

53. © 2013 Pearson Education, Inc. HOMEWORK page 310(12j) Numbers 1 – 23 Skip #9 Review 12A and 12B On 12B (Skip #12) Review 12A and 12B On 12B (Skip #12)