# Physicopoeia

###### φυσικοποιΐα

# Vectors and Scalars

To be honest, the more I teach this subject, the more I find myself thinking it shouldn't really be a big thing in school-level Physics. Teaching the distinction between speed and velocity or displacement and distance __is__ worthwhile, especially when it comes to convincing your pupils that an object moving in a circle at a constant speed __is__ actually accelerating. However, as soon as you move into determining whether other quantities are scalars or vectors, you're entering quite a minefield where things can't always be adequately justified without recourse to some Maths ideas your pupils may well not have encountered. In the meantime, I've seen textbooks (plural) and at least one A-level mark scheme get some of the things below wrong:

ELECTRIC CURRENT

...is a SCALAR.

If you go back to the definition of Current as (Charge ÷ time), Current must be a scalar, since both Charge and time are scalars.

Pupils who have come across Kirchhoff's laws might then ask 'What about when we're considering currents flowing into or out of a junction - surely there's a directionality there that implies it's a vector?' (or words to that effect). On the face of it, this might seem a reasonable objection but, in those situations, it's not really a vector in the proper sense because you only have two choices of direction for the current in a wire. The direction is really just a sign (+ or -) rather than a fully variable direction in the vector sense.

Perhaps the simplest reason why Current is definitely a scalar is because you wouldn't consider a current in a wire as changing direction just because you bend the wire.

ENERGY (of any kind)

...is a SCALAR, as is Work Done.

Whilst this might be obvious in many cases (referring to '38J of Thermal Energy to the left' would not sound sensible in most contexts), it can be less obvious that Work Done, GPE and KE are all scalars as well.

For example, it might be confusing that Force is a vector yet Force x Distance isn't. In fact, the more mathematically concise definition of Work Done is *W* = * F*.

*, as sometimes comes out in post-16 courses where*

__d__*W*=

*F d*cos

*θ*comes up. Since, by definition, the dot product of two vectors is a scalar, Work Done must be a scalar. See also Moment, below.

From GPE = *mgh*, the inclusion of the vectors *g *and *h* can make GPE seem like a vector: moving an object up, down or sideways results in radically different GPE values. However, the mathematically 'proper' version of this equation is GPE = *m g*.

*(*

__h__*i.e.*the dot product of

*and*

__g__*multiplied by*

__h__*m*), which comes directly from

*W*=

*.*

__F__*, above. Again, since dot products by definition give a scalar result, GPE is just as much a scalar as Work Done.*

__d__Similarly, the inclusion of *v* in KE = ½*mv*² also tempts people to think of KE as a vector. Once again, the better version of this equation is ½*m*(* v*.

*) (*

__v__*i.e.*the

*v*² is actually the dot product of

*v*with itself) so the dot product once again produces a scalar result.

PRESSURE

...is a SCALAR.

Given some of the explanation above, it might be tempting to say that Pressure must be a vector because its definition includes Force. Unfortunately, this leads to problems, such as 'what would the direction of the pressure be inside a gas?'. The answer 'in all directions' simply shows that direction is irrelevant.

Mathematically, pressure is defined using d* F* = -

*p*d

*so p is really just a constant of proportionality relating the Force vector to the Area vector.*

__A__'Area vector', I hear you cry? Read on...

AREA

...is a VECTOR.

This may seem silly if you haven't come across it before, but Areas in Physics are usually defined as vectors, where the vector is defined as pointing in a direction perpendicular to the area concerned. Wikipedia has a helpful section on this with diagrams, explaining why pressure must be a scalar.

MOMENT/TORQUE/COUPLE

...are all VECTORS.

Intuitively, this probably makes sense: whether a moment acts in a clockwise or anticlockwise direction clearly makes an enormous difference in most situations. However, it can be confusing that Work Done and Moment seem to have the same definition (Force multiplied by Distance), yet one is a scalar and one a vector. In fact, Work Done is defined as the __dot__ product of F and d but Moment is the cross product of F and d. By definition, cross products give a vector result so Moment is definitely a vector.

You might now be able to see why I'm becoming increasingly less convinced that this has any reasonable place in school-level Physics!