2. Are the quantities that has magnitude only only  Length  Area  Volume  Time  Mass Are quantities that has both magnitude and a direction in space and a direction in space  Displacement  Velocity  Acceleration  Force We say Displacement vector,velocity vector, …………………

3. Physical quantities Scalar Physical quantities الكميات الفيزيائية القياسية only magnitude Vector Physical quantities الكميات الفيزيائية المتجهة temperature Tim Energy Work density magnitude direction displacement Velocity acceleration Force

4. The length of the arrow corresponds to the magnitude of the vector. The direction the arrow points is the vector direction.

5. Examples include: 1- ذيل السهم اتجاه 3- طول السهم يدل على المقدار 1- راس السهم يدل على الاتجاه A vector quantity is a quantity that has both magnitude and a direction in space TailTail Head

7. A B C Suppose that a particle moves from A to B and then later from B to C, we have two displacement vectors AB and BC, The net displacement of these two displacement is single displacement from A to C, we call AC the vector sum. b as a b s sab = +

8. General procedure for adding two vectors graphically:  On paper, sketch vector to some convenient scale and at the proper angle.  Sketch vector to the same scale, with its tail at the head of vector, at the proper angle  The vector sum is the vector that extends from the tail of to the head of.

9. If the vector a and b have the same direction and the same magnitude, we say that vector a equal b and they are parallel.

10. The vector and its opposite

11. Commutative law: Associative law:

12. Vector addition defined in this way,has two important properties a + b = b +a Commutative law ; Adding a to b gives the same result as adding b to a. When there are more than two vectors, we can add vectors a b and c (a + b ) + c = a + (b + c) Associative law

13. Vectors in the same direction Vectors in opposite directions

14. a d c b a+b+c+d Associative law a b a+b c (a+b)+c b b+c a+(b+c) c a Exampl:

15. Homework

17. a) b) The minimum possible magnitude for c The maximum possible magnitude for c a=3m b=4m a=3m b=4m

19. +x +y A AxAx AyAy  The horizontal, or x- component, of A is found by A x = A cos   The vertical, ory-component, of A is found by A y = A sin   Components Vectors * To find the projection of A vector a long an axis we draw perpendicular lines from the two ends of the vector to the axis and it is, AyAy AxAx

20. vector magnitude direction

21. x-x y -y If we were to reverse vector If the vector shifted without changing its direction,its components do not change. then both components would be negative Example

22. X m y m The component of on the x axis is positive, and that on they y axis is negative Angle vector

23. SUMMARY 1. Find the x- and y-components of each vector A x = A cos  A y = A sin  B x = B cos  B y = B sin  C x = C cos  C y = C sin  2. Sum the x- components This is the x-component of the resultant. Rx =Rx = 3. Sum the y- components. This is the y-componentst of the resultant. R y = Pythagorean Theorem 4. Use the Pythagorean Theorem to find the magnitude of the resultant vector. R x 2 + R y 2 = R 2

24. Vector Components:The projection of a vector on an axis is called its component.

28. The unit vectors are dimensionless vectors that point in the direction along a coordinate axis that is chosen to be positive

29. Unit Vectors

30. All vectors can be expressed as a linear combination of these 3 vectors x y z i j i is the unit vector in the x- direction j is the unit vector in the y- direction k is the unit vector in the z- direction k

31. Vector components Scalar components Example

34. Multiplying Vectors Multiplying a Vectors by a scalar Multiplying a Vectors by a vector The Vector product The scalar product Cross productdot product الناتج متجهة الناتج قيمة عددية

35. The Scalar Product : mag. of a mag. of b angle between the vectors

36. Ø=0 b Properties1) a Ø=90 b Ø=180 ba perpendicular

37. 3) Commutative 2)

38. When tow vectors are in unit – vector notation 4) x z y

39. a)

40. c) b)

44. Multiplying Vectors Multiplying a Vectors by a scalar Multiplying a Vectors by a vector The Vector product The scalar product Cross productdot product الناتج متجهة الناتج قيمة عددية

45. Vector Product (aka cross product) The vector product produces a new vector who’s magnitude is given by : The Vector product

46. Ø=0 b Ø=180 a Properties 1) a Ø=90 b perpendicular

47.  Magnitude is  Direction is determined by right-hand rule  Cross production is a vector  Magnitude is

49. Mathe matically, we can find the direction using matrix operations. The cross product is determined from three determinants 4)

50. 3 rd : Strike out the 2 nd column and first row 4 th : Cross multiply the four components,subtract, and multiply by -1: 2 nd : Cross multiply the four components – and subtract: x - component The determinants are used to find the components of the vector 1 st : Strike out the first column and first row! y - component

51. 5 th : Cross out the last column and first row 6 th : Cross multiply and subtract four elements z-component So then the new vector will be:

52. Example :

53. a)

54. b)

55. x z y x z y

56. The Scalar product الضرب القياسي The vector product الضرب الاتجاهي SUMMARY ( ( Dot product cross product

57. If two vectors are parallel or anti-parallel,. If two vectors are perpendicular to each other

58. If and, what is

59. Consider the two vectors represented in the drawing. Which of the following options is the correct way to add graphically vector http://www.wepapers.com/course_view/259/Solutions- _Fundamentals_of_physics_,_Halliday,_Resnicky_8th/Courses/550/

61. بعض المصطلحات الموجودة في Ch3 المصطلح العلمي ( انجليزى ) المصطلح العلمي ( العربي ) Vectorsالمتجهات magnitudeالمقدار Vector quantitiesكميات متجهة Scalar quantitiesكميات قياسية Vectors Propertiesخصائص المتجهات Components Vectorsمركبات المتجهات directionالاتجاه associative lawقانون الجمع Commutative lawقانون التبادل Tailذيل Headراس

62. بعض المصطلحات الموجودة في Ch3 المصطلح العلمي ( انجليزى ) المصطلح العلمي ( العربي ) Vectors المتجهات magnitude المقدار Vector quantities كميات متجهة Scalar quantities كميات قياسية Adding اضافة sum مجموع Vectors Properties خصائص المتجهات Components Vectors مركبات المتجهات Unit Vectors متجهة الوحدة angle زاوية direction الاتجاه IN Subtractions الطرح

63. بعض المصطلحات الموجودة في Ch3 المصطلح العلمي ( انجليزى ) المصطلح العلمي ( العربي ) Vectorsالمتجهات magnitudeالمقدار Vector quantitiesكميات متجهة Scalar quantitiesكميات قياسية perpendicularعمودي sumمجموع Vectors Propertiesخصائص المتجهات Components Vectorsمركبات المتجهات Unit Vectorsمتجهة الوحدة Multiplying Vectorsضرب المتجهات Scalar productالضرب القياسي vector productالضرب الاتجاهي angleزاوية directionالاتجاه Commutative lawقانون التبادل