1. Subtracting Vectors A – B = ?

2. Example Problem What is 30 m/s north minus 15 m/s east?

3. Example Problem What is 30 m/s north minus 15 m/s east? Question: How many vectors are there in this problem?

4. Example Problem What is 30 m/s north minus 15 m/s east? Question: How many vectors are there in this problem?

5. Words  Math What is 30 m/s north minus 15 m/s east?

6. Mathematically A – B = ? Words  Math

7. 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.

8. The Trick: Make it Addition A – B = A + (– B)

9. The Trick: Make it Addition A – B = A + (– B) Guess: what is – B?

10. Time Out! C This is C

11. Time Out! C Point in the direction of – C!

12. Time Out! C Point in the direction of – C!

13. Time Out! C Point in the direction of – C! - C

14. Continue

15. The Trick: Make it Addition A – B = A + (– B) - B is B in the opposite direction

16. The Trick: Make it Addition A – B = A + (– B) - B is B in the opposite direction

17. 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

18. 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)

19. 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)

20. 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

21. 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

22. 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

23. 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

24. 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)

25. 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 (6.70)(5) m/s = 33.5 m/s 6.70 cm

26. 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) 63˚

27. 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) 33.5 m/s @ 63˚ north of west 1 cm = 5 m/s 63˚ Your final answer!

28. Motion Notes (Continued):