1. Vectors and Scalars

2. Scalars
A scalar quantity is a quantity that has magnitude only and has no direction in space
Examples of Scalar Quantities:
Length
Area
Volume
Time
Mass

3. Vectors
A vector quantity is a quantity that has both magnitude and a direction in space
Examples of Vector Quantities:
Displacement
Velocity (speed in a given direction)
Acceleration (rate of which something speeds up or slows down in a direction)
Force

4. Vector Diagrams Vector diagrams are shown using an arrow
Because these two vectors are the same length but are in opposite directions we have u and –u.
Vector diagrams are shown using an arrow
The length of the arrow represents its magnitude
The direction of the arrow shows its direction u v -u
Because these two vectors (u and v) are going in the same direction and have the same magnitude (length) then the vectors are the same u=v.

5. Resultant of Two Vectors
The resultant is the sum or the combined effect of two vector quantities
Vectors in the same direction:
6 N 4 N = 10 N
6 m
= 10 m
4 m
Vectors in opposite directions:
6 m s m s-1 = 4 m s-1
6 N 10 N = 4 N

6. The Parallelogram Law The Triangle Law
When two vectors are joined tail to tail
Complete the parallelogram
The resultant is found by drawing the diagonal
The Triangle Law
When two vectors are joined head to tail
Draw the resultant vector by completing the triangle

7. Adding Vectors “Tip to Tail” method
Suppose that we have the following two vectors
Graphically represent u+v. v v u u
u+v
Resultant vector

8. Graphically represent 2u+3v

9. Represent v-u
Is that the same thing as v+(-u)? v u
v-u -u

10. Sketch u+v, 2(u+v), u-v u v

11. Problem: Resultant of 2 Vectors
Two forces are applied to a body, as shown. What is the magnitude and direction of the resultant force acting on the body?
Solution:
Complete the parallelogram (rectangle)
The diagonal of the parallelogram ac represents the resultant force
The magnitude of the resultant is found using Pythagorean Theorem on the triangle abc
12 N a d θ
5 N
13 N 5 b c 12
Resultant displacement is 13 N 67º with the 5 N force.
Our text book often refers to direction based off of due north going clockwise.

12. Problem: Resultant of 3 Vectors
Find the magnitude (correct to two decimal places) and direction of the resultant of the three forces shown below.
Solution:
Find the resultant of the two 5 N forces first (do right angles first) 5 d c
7.07 N
Now find the resultant of the 10 N and N forces
5 N 5
90º
The 2 forces are in a straight line (45º º = 180º) and in opposite directions
45º θ a
2.93 N b
5 N
135º
So, Resultant = 10 N – 7.07 N = 2.93 N in the direction of the 10 N force
10 N

13. Recap What is a scalar quantity? Give 2 examples
What is a vector quantity?
How are vectors represented?
What is the resultant of 2 vector quantities?
What is the triangle law?
What is the parallelogram law?

14. Resolving a Vector Into Perpendicular Components
When resolving a vector into components we are doing the opposite to finding the resultant
We usually resolve a vector into components that are perpendicular to each other
Here a vector v is resolved into an x component and a y component v y x

15. Practical Applications
Here we see a table being pulled by a force of 50 N at a 30º angle to the horizontal
When resolved we see that this is the same as pulling the table up with a force of 25 N and pulling it horizontally with a force of 43.3 N
50 N
y=25 N
30º
x=43.3 N
We can see that it would be more efficient to pull the table with a horizontal force of 50 N

16. Calculating the Magnitude of the Perpendicular Components
If a vector of magnitude v and makes an angle θ with the horizontal then the magnitude of the components are:
x = v Cos θ
y = v Sin θ v
y=v Sin θ y θ
x=v Cos θ x
Proof:

17. Problem: Calculating the magnitude of perpendicular components
2002 HL Sample Paper Section B Q5 (a) A force of 15 N acts on a box as shown. What is the horizontal component of the force?
Solution:
12.99 N
15 N
Component
Vertical
60º
Horizontal
Component
7.5 N

18. 2003 HL Section B Q6 Solution:
A person in a wheelchair is moving up a ramp at constant speed. Their total weight is 900 N. The ramp makes an angle of 10º with the horizontal. Calculate the force required to keep the wheelchair moving at constant speed up the ramp. (You may ignore the effects of friction).
Solution:
If the wheelchair is moving at constant speed (no acceleration), then the force that moves it up the ramp must be the same as the component of it’s weight parallel to the ramp.
Complete the parallelogram.
Component of weight
parallel to ramp: N
10º
80º
10º
Component of weight perpendicular to ramp: N
900 N

19. Summary
If a vector of magnitude v has two perpendicular components x and y, and v makes and angle θ with the x component then the magnitude of the components are:
x= v Cos θ
y= v Sin θ v
y=v Sin θ y θ
x=v Cosθ