# Vectors and the addition of vectors

If you want to row straight across a river when a current is flowing you must point the bow slightly upstream
A pilot must allow for the wind speed and direction when piloting an aircraft
A horse pulling a barge along a canal is more effective if the rope is long
Rugby forwards are more effective if they push the scrum from behind
A kicker in rugby or soccer must allow for the wind speed and direction when taking a place kick

All these facts are connected with vectors and the addition of vectors.
Vectors are quantities that have direction as well as size. Example of vectors are weight, force and velocity.
Quantities with no direction are called scalars and examples of these are mass, length and energy.

If you pull on a truck that is mounted on rails the truck will move along the rails. However just how quickly it will move will depend not only on how hard you pull but also on the direction in which you pull.

Obviously if you pull in the direction of the rails the truck will move quickly (a). If you pull at an angle to the track it will move but not so quickly (b), but if you pull at right angles to the rails (c) the truck will not move at all (unless you actually pull it sideways off the track!).

Now imagine that you and a friend want to pull a box along the road. (To make life easy lets imagine that the road is smooth – no friction). The box has two ropes fixed to it and you each take hold of one of them.

Which way should you pull?

If you both pull in the same direction with the same force the box will accelerate quickly (d) but if you pull in opposite directions with the same force you would expect the box to stay still and this is what would happen (e).

If you were stronger than your friend then the box would move the way you were pulling but not as fast if you pulled on your own (f).

The red arrows show the effect of both forces together.

What if one of you pulls at an angle to the other, the box would move off in a direction somewhere between the two ropes (g) and with less acceleration than if you both pulled in the same direction – the effective total force would be less. The simplest case is where the ropes are at right angles so we will look at this case first.

There are two ways of working out exactly what the total force would be in each case:
(a) by a scale drawing
(b) by mathematics and trigonometry (Pythagoras).

(a) Think of the two forces acting one after another. Draw them in a scale diagram nose to tail (blue arrows) using the length of the line to represent the size of the force. The combined effect (resultant) of these two forces is then the line (red) that completes the triangle.
You can find the size and direction of the resultant simply by measuring your diagram.

(b) Using Pythagoras' theorem you can calculate the size of the hypotenuse of the triangle and then use trigonometry to find the direction of the resultant.

Example problems
Two tractors try to rescue a cow that is trapped in a ditch. One pulls along the ditch with a force of 500 N and the other pulls at right angles to the ditch with a force of 250 N. What is the combined force on the cow and in which direction will the cow move?

Use a scale diagram showing the two forces and the resultant.

Measuring the diagram shows that the resultant force on the cow is just under 560 N and that it is at an angle of 27o with the ditch.

If a boat is sailing across a lake then you can work out how long it will take if you know the speed of the boat and the distance across the lake. This is fine if the water is still.

However what happens if the water is moving as it is in a river. The speed of the boat compared with the bank of the river then depends not only on its speed relative to the water but also on the sped of the water itself.

We should really be talking about velocity here because if you remember velocity is a vector and vectors are quantities that depend on direction as well as size.

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