# Vectors and scalars

Physics Narrative for 14-16

## Some quantities have direction: some do not

Some physical quantities are essentially directional. Force is one such. As soon as you describe a force, it's possible to ask about its direction. You can't answer that it has no particular direction, or any direction at all. These quantities are described using vectors – ordered sets of numbers, together with units.

Some physical quantities have no direction – that is, it makes no sense to attach a direction to the description. Energy is one such quantity. Such quantities are described with a single number, and usually a unit.

Descriptions of motion are best treated with vectors, and that is what we've suggested throughout; →a, →v, →s. These are all properties of an object (as seen from a chosen point of view), so it makes sense for the arrows to be linked to such an object, just as forces always act on an object.

Simplifications to a single dimension can lead to acceleration, velocity and position being treated as signed quantities. Such treatments can obscure the essential vector nature of the entities, and create difficulties (more on this in the Teaching and Learning Issues strand).

How many ordered numbers you need to define the quantity depends on the number of dimensions. In this world we normally use three dimensions.

→v = 3 metre / second2 metre / second − 1 metre / second

→s = − 5 metre2 metre − 7 metre

## Distance, position and displacement

Displacement is a vector quantity that refers to "how far out of place an object is"; it is the object's overall change in position.

That suggests that you ought to be rather careful to separate moving things from repositioning them, perhaps reserving moving

for changes where you're interested in the journey, and not only in the end points.

If you are interested only in the end points then changing the position is equivalent to displacing the object.

Distance is a scalar quantity that refers to how much ground an object has covered

during its motion.

If you undertake a trip or journey then you move – that is you traverse a distance. This may involve lots of wiggles, so the distance may not be equal to the magnitude of the displacement (||→s ||).

## Acceleration, velocity, speed

Acceleration is a vector, so it needs an ordered set to describe it. One such set can include the magnitude of the acceleration. Other members of the set will then have to include the direction of the acceleration. As the world is three-dimensional, there will be three numbers, in this case the magnitude and two angles.

→a = magnitudeangleangle

You'd have to already have a pair of lines defined, from which to measure the angles.

You could use three other numbers. For example:

→a = acceleration in x directionacceleration in y directionacceleration in z direction

Again, here you need some agreed background to make this description intelligible – here the direction of the x, y and z axes.

In summary, →a is useful, and ||→a || can be useful, but there's no special name in general use.

Velocity is a vector, so it's best described by an ordered set of numbers and units. One such set can include the magnitude of the velocity. Other members of the set will then have to include the direction of the velocity. As the world is three-dimensional, there will be three numbers, in this case the magnitude and two angles.

→v = magnitudeangleangle

You'd have to already have a pair of lines defined, from which to measure the angles.

You could use three other numbers. For example:

→v = velocity in x directionvelocity in y directionvelocity in z direction

Again, here you need some agreed background to make this description intelligible – here the direction of the x, y and z axes.

In summary, →v is useful, and ||→v || can be useful, and it has an everyday name: speed.