Vectors & Scalars

Vectors & Scalars

It would not make sense to ask a grocer for 2 kg of potatoes due west. Masses don't have direction: they just have size or 'magnitude'. Quantities having only magnitude and no direction are called scalars.

On the other hand, it does make sense to say that the Earth exerts a force on the said potatoes of 20 N downwards. Forces do have a direction. Quantities having direction as well as magnitude are called vectors.

Here is a table of some common scalars and vectors.

Scalars	Vectors	Notes
Mass	Weight	Subject of trick questions. Mass is in kg. Weight is in N and calculated from mass by multiplying by g (which is a vector)
	Force
Time		People sometimes maintain that time has a direction because it runs forwards rather than backwards. This is using direction in a quite different sense. Time is a scalar because you couldn't say the time is 10 minutes due north.
Speed	Velocity	Subject of trick questions. Velocity is always a vector really (in that there invariably is a direction if something is moving), but if you aren't told the direction, or are not interested in it (e.g. when discussing the world land speed record), then you call it speed. If something is moving in a circle, it might have a constant speed even though its velocity is changing all the time by virtue of its direction changing all the time.
Pressure		In the equation Force = Pressure x Area, it is the Area that is regarded as the vector, its direction being perpendicular to the area itself.
Energy
	Acceleration	You might not be told the direction, in which case you treat it as a scalar: but it's still called acceleration.
	Momentum	You might not be told the direction, in which case you treat it as a scalar: but it's still called momentum.

Representation of vectors

Weather maps sometimes show high winds as fat arrows and low winds as slender arrows. Sometimes high winds are shown as lots of closely spaced arrows. A physicist would depict a high wind as a long arrow. We make the length of the arrow correspond to the magnitude, using some scale (which we always declare before we start), and we make the direction of the arrow correspond to the direction of the velocity/force/momentum/etc relative to some baseline (which we always declare before we start).

Note that weight, for example, possesses location as well as magnitude and direction. It acts through the centre of mass of the relevant body rather than at some random point in space. When we are depicting forces on diagrams, we must always be careful to take account of location and ensure that the back end of the arrow starts on the point at which the force acts. But when we draw vector diagrams we ignore location and feel free to slide the arrows around the page, just being careful to do so in such a way that the arrow always remains parallel to its original self.

Addition of vectors

Draw a diagram of the physical situation. Declare a magnitude scale and a baseline for directions (or compass rose)
Draw vector arrows in the right place, of the right length and in the right direction - i.e. taking location into account.
Re-draw the vectors in a blank part of the paper, but preserving their length and orientation. Draw them in such a way that each new vector begins where the last one left off. This produces a polygon (often a triangle) with one side missing. The resultant vector (the vector sum) is found by going back to the beginning of the first vector in the chain and drawing a line directly to the tip of the last vector in the chain.
All of this is particularly simple to do if the two vectors are in the same (or opposite) directions.
Sometimes the vector polygon will appear not to have one side missing. This is because the missing side is of zero length, which, in turn, is because the tip of the final vector comes back to join up with the start of the first vector. Such a system is in equilibrium: there is no resultant force acting , and there is no acceleration.