Introduction to Vectors in Motion

1. Chapter 2 Motion in two dimensions 2.1: An introduction to vectors

2. 2.1: An introduction to vectors • 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

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: Two vectors can be added using these method: 1- tip to tail method. 2- the parallelogram method. 1- tip to tail method. Graphical method (triangle method): 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.

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

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 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 q between vector and x-axis:

12. The signs of the components Ax and Ay depend on the angle q 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. 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 The unit vector notation for the vector A is: OR in even better shorthand notation:

15. Adding Vectors by Components We want to calculate: R = A + B From diagram: R = (Axi + Ayj) + (Bxi + Byj) R = (Ax + Bx)i + (Ay + By)j Rx = Ax+ Bx Ry = Ay+ By The components of R: The magnitude of a R: The angle q 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 and the positive x axis is: • 61° • 29° • 151° • 209° • 241°

19. Example :

22. W Example : F1 = 37N 54° N of E F2 = 50N 18° N of F3 = 67 N 4° W of S F=F1+F2+F3

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