1. 2.1: An introduction to vectors
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: 1- tip to tail method.
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. A + B = B + A Commutative Law of vector addition
2- the parallelogram method.
A + B = B + A
(Parallelogram rule of addition)

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

8. A - B = A + (-B) Subtracting vectors: 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. F1 = 37N 54° N of E F2 = 50N 18° N of F3 = 67 N 4° W of S
Example :
F1 = 37N 54° N of E F2 = 50N 18° N of F3 = 67 N 4° W of S
F=F1+F2+F3 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