Scalars and VectorsScalar quantities have only magnitude (size) but vector quantities have magnitude and direction. For example, the weight (vector) of an object acts down towards the Earth's centre, but its mass (scalar) is just a quantity  it has no direction. Forces (vectors) can push in different directions  but energy (scalar) has no direction associated with it. Confusion often arises when we use a scalar term and/or vector term in everyday life as 'interchangable'. Speed is a scalar quantity and velocity is a vector quantity. Consider the following example:
It travels from A to B around the track. The distance it travels is therefore half of the circumference. But its displacement (vector version of distance) after half a lap is the diameter of the circle. speed = distance travelled/time taken = πd/(2t)
velocity magnitude = d/t You represent vectors with an arrow:the length of the arrow represents the magnitude of the vector and the direction it points on the page represents the direction. You usually put a little arrow over the top of the letter representing a vector in your diagrams.
Vector AdditionTo add scalars is easy  you just add the values together, but adding vectors is a little more complicated. You have to take the direction into account. At GCSE you will only have to add colinear vectors (ones that are in a straight line). Vectors pulling in the opposite direction simply cancel the effect of the each other out. At A Level it is a little more complicated. You have to tackle vectors that point in all directions!
When drawing out vectors you have to be careful to draw them very carefully.
Measuring the 'angle of a vector'
Vector addition by calculationTo add vectors that are pointing in different directions you need to break them down into their horizontal and vertical components. These can then be easily added together to obtain the vertical and horizontal components of the resultant vector. Simple use of Pythagoras Theorem then gives you the magnitude of the vector  and a little bit of 'trig' gets you the angle. It is really simple to do!
Subtracting vectorsTo subtract a vector simple reverse its direction  then add it! Multiplying vectors by scalarsThis affects the magnitude only. Multiplying vectors by vectors (cross and dot products)This can affect magnitude and/or direction  it is not required at A level in Physics  you will do that at University. PracticeYou can practise resolving vectors into their components by using my interactive XL spreadsheet. Try it out! Simply make up a vector (say, 64N at 23 degrees to the horizontal) and then use your calculator to work out the horizontal and vertical components of it. The spreadsheet will tell you if you have done it correctly. Enter the values into the sheet and it will work out the components for you. This is a useful tool for you to use when you need extra practice. Resolving vectors on an inclined plane has its own page!
Here is a link to my page on three coplanar forces in equilibrium.... and here is an interactive applet...

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