1. Vector & Scalar Quantities

2. Characteristics of a Scalar Quantity
Only has magnitude
Requires 2 things:
1. A value
2. Appropriate units
Ex. Mass: 5kg
Temp: 21° C
Speed: 65 mph

3. Characteristics of a Vector Quantity
Has magnitude & direction
Requires 3 things:
1. A value
2. Appropriate units
3. A direction!
Ex. Acceleration: 9.8 m/s2 down
Velocity: mph West

4. More about Vectors A vector is represented on paper by an arrow
1. the length represents magnitude
2. the arrow faces the direction of motion
3. a vector can be “picked up” and moved on
the paper as long as the length and direction
its pointing does not change

5. Graphical Representation of a Vector
The goal is to draw a mini version of the vectors to give you an accurate picture of the magnitude and direction. To do so, you must:
Pick a scale to represent the vectors. Make it simple yet appropriate.
Draw the tip of the vector as an arrow pointing in the appropriate direction.
Use a ruler & protractor to draw arrows for accuracy. The angle is always measured from the horizontal or vertical.

6. Understanding Vector Directions
To accurately draw a given vector, start at the second direction and move the given degrees to the first direction. N
30° N of E W E
Start on the East origin and turn 30° to the North S

7. Graphical Representation Practice
5.0 m/s East
(suggested scale: 1 cm = 1 m/s)
300 Newtons 60° South of East
(suggested scale: 1 cm = 100 N)
0.40 m 25° East of North
(suggested scale: 5 cm = 0.1 m)

8. Graphical Addition of Vectors
Tip-To-Tail Method
Pick appropriate scale, write it down.
Use a ruler & protractor, draw 1st vector to scale in appropriate direction, label.
Start at tip of 1st vector, draw 2nd vector to scale, label.
Connect the vectors starting at the tail end of the 1st and ending with the tip of the last vector This = sum of the original vectors, its called the resultant vector.

9. Graphical Addition of Vectors (cont.)
Tip-To-Tail Method
5. Measure the magnitude of R.V. with a ruler. Use your scale and convert this length to its actual amt. and record with units.
6. Measure the direction of R.V. with a protractor and add this value along with the direction after the magnitude.

10. Tip-to-Tail Method

11. Graphical Addition of Vectors (cont.)
5 Km
Scale: 1 Km = 1 cm
3 Km
Resultant Vector (red) = 6 cm, therefore its 6 km.

12. Vector Addition Example #1
Use a graphical representation to solve the following: A hiker walks 1 km west, then 2 km south, then 3 km west. What is the sum of his distance traveled using a graphical representation?

13. Vector Addition Example #1 (cont.)
Answer = ????????

14. Vector Addition Example #2
Use a graphical representation to solve the following: Another hiker walks 2 km south and 4 km west. What is the sum of her distance traveled using a graphical representation? How does it compare to hiker #1?

15. Vector Addition Example #2 (cont.)
Answer = ????????

16. Mathematical Addition of Vectors
Vectors in the same direction:
Add the 2 magnitudes, keep the direction the same.
Ex =
3m E m E m E

17. Mathematical Addition of Vectors
Vectors in opposite directions
Subtract the 2 magnitudes, direction is the
same as the greater vector.
Ex.
4m S m N = m S

18. Mathematical Addition of Vectors
Vectors that meet at 90°
Resultant vector will be hypotenuse of a
right triangle. Use trig functions and
Pythagorean Theorem.

19. Addition of Vectors (contd.)b c c b b a a
c = a + b
Parallelogram Law

20. Addition of Vectors (contd.)
Head-to-tail c d b a b c e d e a f
f = a + b + c + d +e

21. Mathematical Subtraction of Vectors
Subtraction of vectors is actually the addition of a negative vector.
The negative of a vector has the same magnitude, but in the 180° opposite direction.
Ex. 8.0 N due East and 8.0 N due West
3.0 m/s 20° S of E and 3.0 m/s 20° N of W

22. Subtraction of Vectors (cont.)
Subtraction used when trying to find a change in a quantity.
Equations to remember:
∆d = df – di or ∆v = vf – vi
Therefore, you add the second vector to the opposite of the first vector.

23. Subtraction of Vectors (cont.)
Ex. = Vector #1: 5 km East
Vector #2: 4 km North
5 km W (-v1)
4 km N (v2)

24. y1 y2 ay x1 x2 ax Look a two dimensional vector in a
2D Cartesian Coordinates
Look a two dimensional vector in a
2D Cartesian Coordinate System O X Y y1 y2 a  ay x1 x2  ax

25. 2D Cartesian Coordinates (contd.)Y O X a ax ay  
a =  ax ay2
ax=x2 –x1= a cos 
ay=y2 –y1= a cos  ay ax

26. Vector Components Resolving a vector : The process of finding the
components of the vector.
Coordinate System

27. The component of the vector along an axis is
Component of a vector
The component of the vector
along an axis is
its projection along that axis O x y a   ax ay

28. Component Method (cont.)
4. Add all the “X” components (Rx)
5. Add all the “Y” components (Ry)
6. The magnitude of the Resultant Vector is
found by using Rx, Ry & the Pythagorean
Theorem:
R2 = Rx2 + Ry2
7. To find direction: Tan Θ = Ry / Rx

29. Component Method (cont.)
Ex. #1
V1 = 2 m/s 30° N of E
V2 = 3 m/s 40° N of W
Find: Magnitude & Direction
Magnitude = 2.96 m/s
Direction = 78° N of W

30. Component Method (cont.)
Ex. #2
F1 = 37N 54° N of E
F2 = 50N 18° N of W
F3 = 67 N 4° W of S
Find: Magnitude & Direction
Magnitude =37.3 N
Direction = 35° S of W