1. Adding Vectors
E.g. A boat is pulled into harbour by two tug boats at right angles as shown in the diagram – calculate its resultant speed
NOTE – because they are vectors AND not in the same direction we can’t simply write down 5 ms-1
3 ms -1
Instead we make a triangle out of the two velocities and use Pythagoras
2 ms -1

2. Triangle of forces
Make the two forces that we want to add together into a triangle NOSE to TAIL. Like this …
2 ms -1
3 ms -1 OR
3 ms -1
2 ms -1
It’s a right angled triangle so use Pythagoras:
Hypotenuse2 = =
= 13
Result =  13 = 3.6 ms-1
The result of adding these two vectors is the missing side of the triangle.
Its length will be the speed of the ship and its direction will be as drawn.

3. Subtracting Vectors
E.g. subtract Force B from Force A (which are at right angles) … A
It is the same as adding – but point the subtracting force in the other direction A -B B
Then nose to tail them, and do Pythagoras

4. Vectors that are not at right angles
Draw a scale drawing and then a parallelogram around the two forces – the resultant is the diagonal:
3 ms -1
300
Horizontal
250
2 ms -1

5. Occasionally you can resolve
500
horizontal
Special case:
The forces are equal and at the same angle – therefore the resultant will be horizontal – therefore resolve both forces to the horizontal and add them
500
Result = 2 x 10 cos 50
10N
Note – Pythagoras won’t work in this case – why not?