Vectors in Coordinate Systems

1. المحاضرة الثانية والثالثةVectors

2. 3.1 Coordinate Systems 1- Cartesian coordinates (rectangular coordinates). (x, y) 2- polar coordinate system. (r is the distance an θ is the angle)

3. we can obtain the Cartesian coordinates by using the equations: positive θis an angle measured counterclockwise from the positive x axis

4. Example : Polar Coordinates Note that you must use the signs of x and y to find θ

5. Problem (1)

6. Problem (2)

7. Problem (5)

8. Exercises Problems: (4,6)

9. 3.2 Vector and Scalar Quantities A scalar quantity is completely specified by a single value with an appropriate unit and has no direction. Examples: (volume, mass, speed, distance, time intervals and Temperature).

10. A vector quantity ( or A ) is completely specified by a number and appropriate units plus a direction. Examples: (Displacement , velocity, acceleration and area)

11. The magnitude of the vector A is: 1- written either A or 2- has physical units. 3- is alwaysa positive number.

12. Quick Quiz: Which of the following are vector quantities? (a) your age (b) acceleration (c) velocity (d) speed (e) mass

13. 3.3 Some Properties of Vectors •Equality of Two Vectors A = B only if A = Band if A and B point in the same direction along parallel lines.

14. Example: These four vectors are equal because they have equal lengths and point in the same direction.

15. • Adding Vectors There are three Methods to Adding vectors: 1- Graphical methods 2- Algebraically Method. 3- Vector Analysis Adding vectors= finding the resultant (R)

16. 1- Graphical methods

17. When vector B is added to vector A, the resultant Ris the vector that runs from the tail of A to the tip of B. A R=A+B B B A

18. Example: If you walked 3.0 m toward the east and then 4.0 m toward the north, the resultant displacement is 5.0 m, at an angle of 53° north of east.

19. Geometric construction is Used to add more than two vectors The resultant vector R = A + B + C + D is the vector that completes the polygon

20. R is the vector drawn from the tail of the first vector to the tip of the last vector.

21. commutative law of addition: 1- A + B R

22. 2- B+A R

23. Conclusion (1):

24. Keep in mind that: A + B = C is very different from A + B = C. * A + B = C is a vector sum. * A + B = C is a simple algebraic addition of numbers

25. Associative law of addition: 1- (A + B) + C (A+B)+C (A+B) B

26. 2- A+(B+C) A+(B+C) (B + C) B

27. Conclusion (2):

28. Keep in mind that: When two or more vectors are added together, all of them must have the same units and all of them must be the same type of quantity.

29. Example: A is a velocity vector B is a displacement vector Find : A+B Answer: A+Bhasno physical meaning A B

30. Negative of a Vector The vectors A and -A have the same magnitude but point in opposite directions. A A + (-A) = 0. A -

31. Subtracting Vectors B Find (A-B) A A A-B -B Or B -B A - B A A - B = A + (-B)

33. Quick Quiz The magnitudes of two vectors A and B are A = 12 units and B = 8 units. Which of the following pairs of numbers represents the largest and smallest possible values for the magnitude of the resultant vector R = A + B? (a) 14.4 units, 4 units (b) 12 units, 8 units (c) 20 units, 4 units (d) none of these answers.

34. Quick Quiz If vector B is added to vector A, under what condition does the resultant vector A + B have magnitude A + B? (a) A and B are parallel and in the same direction (b) A and B are parallel and in opposite directions (c) A and B are perpendicular.

35. Quick Quiz If vector B is added to vector A, which two of the following choices must be true in order for the resultant vector to be equal to zero? (a) A and B are parallel and in the same direction. (b) A and B are parallel and in opposite directions. (c) A and B have the same magnitude. (d) A and B are perpendicular.

36. Problem(8) Answer:

37. Problem (10) Answer:

38. Multiplying a Vector by a Scalar - If vector A is multiplied by a positive scalar quantity m, then the product mA is a vector that has the same direction as A and magnitude mA. - If vector A is multiplied by a negative scalar quantity -m, then the product -mA is directed opposite A.

39. Example: The vector 5A is five times as long as A and points in the same direction as A.

40. Exercises Problems: (9,12,14,15,16)

41. 2- Algebraically Method.

42. - The magnitude of R can be obtained from: : is an angle between A and B -The direction of R measured can be obtained from the law of sines:

43. Example : A car travels 20.0 km due north and then 35.0 km in a direction 60.0° west of north, as shown in Figure. Find the magnitude and direction of the car’s resultant displacement. Solution 60 A B

44. Graphical method for finding the resultant displacement vector R = A +B. (in the previous example)

45. 3- Vector Analysis

46. Components of a Vector and Unit Vectors Note thatthe signs of the components Ax and Ay depend on the angle θ.

47. The signs of the components of a vector A depend on the quadrant in which the vector is Located.

48. The magnitude and direction of A are related to its components through the expressions:

49. Quick Quiz Choose the correct response to make the sentence true: A component of a vector is (a) always, (b) never, or (c) sometimes larger than the magnitude of the vector.

50. Example: The components of B along the x' and y' axes are: Bx' = Bcos θ' By' = Bsin θ'