1. Physics and Physical Measurement
Topic 1.3 Scalars and Vectors

2. Scalars Quantities
Scalars can be completely described by magnitude (size)
Scalars can be added algebraically
They are expressed as positive or negative numbers and a unit
examples include :- mass, electric charge, distance, speed, energy

3. Vector Quantities
Vectors need both a magnitude and a direction to describe them (also a point of application)
When expressing vectors as a symbol, you need to adopt a recognized notation
e.g.
They need to be added, subtracted and multiplied in a special way
Examples :- velocity, weight, acceleration, displacement, momentum, force

4. Addition and Subtraction
The Resultant (Net) is the result vector that comes from adding or subtracting a number of vectors
If vectors have the same or opposite directions the addition can be done simply
same direction : add
opposite direction : subtract

5. Co-planar vectors
The addition of co-planar vectors that do not have the same or opposite direction can be solved by using scale drawings to get an accurate resultant
Or if an estimation is required, they can be drawn roughly
or by Pythagoras’ theorem and trigonometry
Vectors can be represented by a straight line segment with an arrow at the end

6. Triangle of Vectors
Two vectors are added by drawing to scale and with the correct direction the two vectors with the tail of one at the tip of the other.
The resultant vector is the third side of the triangle and the arrow head points in the direction from the ‘free’ tail to the ‘free’ tip

7. Example
R = a + b a + b =

8. Parallelogram of Vectors
Place the two vectors tail to tail, to scale and with the correct directions
Then complete the parallelogram
The diagonal starting where the two tails meet and finishing where the two arrows meet becomes the resultant vector

9. Example
R = a + b a + b =

10. More than 2
If there are more than 2 co-planar vectors to be added, place them all head to tail to form polygon when the resultant is drawn from the ‘free’ tail to the ‘free’ tip.
Notice that the order doesn’t matter!

11. Subtraction of Vectors
To subtract a vector, you reverse the direction of that vector to get the negative of it
Then you simply add that vector

12. Example a - b =
R = a + (- b) -b

13. Multiplying Scalars
Scalars are multiplied and divided in the normal algebraic manner
Do not forget units!
5m / 2s = 2.5 ms-1
2kW x 3h = 6 kWh (kilowatt-hours)

14. Multiplying Vectors
A vector multiplied by a scalar gives a vector with the same direction as the vector and magnitude equal to the product of the scalar and a vector magnitude
A vector divided by a scalar gives a vector with same direction as the vector and magnitude equal to the vector magnitude divided by the scalar
You don’t need to be able to multiply a vector by another vector

15. Resolving Vectors
The process of finding the Components of vectors is called Resolving vectors
Just as 2 vectors can be added to give a resultant, a single vector can be split into 2 components or parts

16. The Rule A vector can be split into two perpendicular components
These could be the vertical and horizontal components
Vertical component
Horizontal component

17. Or parallel to and perpendicular to an inclined plane

18. These vertical and horizontal components could be the vertical and horizontal components of velocity for projectile motion
Or the forces perpendicular to and along an inclined plane

19. Doing the TrigonometryV
Sin  = opp/hyp = y/V y
Therefore y = Vsin 
In this case this is the vertical component  x
Cos  = adj/hyp = x/V 
V cos 
V sin 
Therefore x = Vcos 
In this case this is the horizontal component

20. Quick Way If you resolve through the angle it is cos
If you resolve ‘not’ through the angle it is
sin

21. Adding 2 or More Vectors by Components
First resolve into components (making sure that all are in the same 2 directions)
Then add the components in each of the 2 directions
Recombine them into a resultant vector
This will involve using Pythagoras´ theorem

22. Question
Three strings are attached to a small metal ring. 2 of the strings make an angle of 70o and each is pulled with a force of 7N.
What force must be applied to the 3rd string to keep the ring stationary?

23. Answer Draw a diagram 7 cos 35o + 7 cos 35o 7N F 70o 7 sin 35o

24. Horizontally
7 sin 35o - 7 sin 35o = 0
Vertically
7 cos 35o + 7 cos 35o = F
F = 11.5N
And at what angle?
145o to one of the strings.