2. Vectors: Magnitude and direction Examples for Vectors: force – acceleration- displacement Scalars: Only Magnitude A scalar quantity has a single value with an appropriate unit and has no direction. Examples for Scalars: mass- speed- work-Distance- Energy-Work- Pressure Motion of a particle from A to B along an arbitrary path (dotted line). Displacement is a vector 2.1: An introduction to vectors

3. Vectors: Represented by arrows ( example displacement ). Tip points away from the starting point. Length of the arrow represents the magnitude In text: a vector is often represented in bold face (A) or by an arrow over the letter. In text: Magnitude is written as A or This four vectors are equal because they have the same magnitude and same length

4. Adding vectors: Draw vector A. Draw vector B starting at the tip of vector A. The resultant vector R = A + B is drawn from the tail of A to the tip of B. Graphical method (triangle method): Two vectors can be added using these method: 1- tip to tail method. 2- the parallelogram method. 1- tip to tail method.

5. 5 Adding several vectors together. Resultant vector R=A+B+C+D is drawn from the tail of the first vector to the tip of the last vector.

6. 6 A + B = B + A (Parallelogram rule of addition) Commutative Law of vector addition 2- the parallelogram method.

7. Associative Law of vector addition A+(B+C) = (A+B)+C The order in which vectors are added together does not matter.

8. Negative of a vector. The vectors A and –A have the same magnitude but opposite directions. A + (-A) = 0 A -A-A Subtracting vectors: A - B = A + (-B)

9. Multiplying a vector by a scalar The product mA is a vector that has the same direction as A and magnitude mA. The product –mA is a vector that has the opposite direction of A and magnitude mA. Examples: 5A;-1/3A Given, what is ?

11. Components of a vector The x- and y-components of a vector: The magnitude of a vector: The angle  between vector and x-axis:

12. The signs of the components A x and A y depend on the angle  and they can be positive or negative.  Examples)

13. Unit vectors A unit vector is a dimensionless vector having a magnitude 1. Unit vectors are used to indicate a direction. i, j, k represent unit vectors along the x-, y- and z- direction i, j, k form a right-handed coordinate system

14. The unit vector notation for the vector A is: OR in even better shorthand notation: A unit vector is a dimensionless vector having a magnitude 1. Unit vectors are used to indicate a direction. i, j, k represent unit vectors along the x-, y- and z- direction i, j, k form a right-handed coordinate system

15. Adding Vectors by Components We want to calculate:R = A + B From diagram:R = (A x i + A y j) + (B x i + B y j) R = (A x + B x )i + (A y + By)j The components of R: R x = A x + B x R y = A y + B y The magnitude of a R: The angle  between vector R and x-axis:

16. example

17. Example  A force of 800 N is exerted on a bolt A as show in Figure (a). Determine the horizontal and vertical components of the force. The vector components of F are thus, and we can write F in the form

18. Example : The angle between where with the positive x is: 1.61° 2.29° 3.151° 4.209° 5.241°

19. Example :

22. F 1 = 37N 54° N of E F 2 = 50N 18° N of F 3 = 67 N 4° W of S F=F 1 +F 2 +F 3 W

23. Ex : 2 – 10 A woman walks 10 Km north, turns toward the north west, and walks 5 Km further. What is her final position?

24. example Answer is d