1. Dealing with Vectors
Forces are used as an example though rules apply to any vector quantities

2. Representing vectors Vectors can be shown either by:
A line with a length proportional to the magnitude of the vector and a direction shown by the direction of the line
A number with a sign / bearing / direction / angle
(and a unit!)

3. Adding vectors
The resultant of any number of vectors (forces) acting at a point can be found using two methods:
By scale drawing
By calculation
The resultant vector is the one single vector which can replace all the other vectors acting at a point

4. Example 8N 5N
Resultant = = -3N ie 3N to left
The resultant is the sum of the two vectors (remembering about directions)

5. Scale diagrams
The resultant of 2 or more vectors can be found by joining the vectors end to end (keeping their direction correct)
The resultant magnitude and direction is given by the vector joining the starting point to the end point

6. Example 1 8N 5N
This works for any number of vectors, acting at the same point – even ones at an angle 3N 5N 8N

7. Example 2
It doesn’t matter which vector is started with, the magnitude and direction of the resultant is the same

8. So that means…… If all these forces were acting at one point,
then this system of forces would have no resultant because the forces join up end to end.
These forces are in equilibrium

9. Perpendicular pairs of vectors
resultant
Magnitude and direction of the resultant can be calculated using Pythagoras
Complete the rectangle and find the length and direction of the diagonal

10. It also works for….
Pairs of vectors not at right angles - complete the parallelogram to give the same resultant
obviously Pythagoras cannot be used in this case or

11. Remember…. Rules apply in three dimensions
Problems you are given will only be in two dimensions – ie vectors on the same plain
Known as co-planar vectors

12. Scale diagrams Sharp pencil Straight lines using an easy scale
Don’t scale the angles!
Careful angle measurement
Diagrams as large as possible to reduce measurement errors
It doesn’t matter if the vectors cross over

13. Now try one!
4 forces act at a point. Their magnitude and bearings are shown below. By scale diagram, find the resultant force.
F1 = 45N bearing 0 degrees
F2 = 20N bearing 45 degrees
F3 = 55N bearing 150 degrees
F4 = 30N bearing 270 degrees

14. Perpendicular vectors
The resultant of perpendicular vectors can be calculated using Pythagoras.
Find the resultant of these 4 forces 5N 5N 8N 1N

15. Working backwards
Because we can find the resultant of perpendicular vectors, then if we can turn non-perpendicular vectors into a series of perpendicular ones, then we can find their resultant.
Can be redrawn as 2 perpendicular vectors

16. Using trig.
Sin, cos and tan relationships can be used to work out the lengths / magnitudes of each side of the triangle
F sin θ F θ
F cos θ

17. Find the resultant of this combination of forces
The resultant of perpendicular vectors can be calculated using Pythagoras.
Find the resultant of these 4 forces 8N 5N 5N
37 degrees 1N