A force is not just a vector

You’ll often read (textbook, webpage etc.) or hear (teacher, student etc.), the phrase “A force is a vector quantity”… or, for short, “A force is a vector”. For example, if you google “a force is a vector” the first reference (as of writing) is The Meaning of Force by the Physics Classroom. Their presentation is very similar to many others (I’m not picking on them, it was just the easiest reference to find… as I often like to point out, physicists are usually very lazy efficient). The lesson states (my bold):

A force is a vector quantity… a vector quantity is a quantity that has both magnitude and direction. To fully describe the force acting upon an object, you must describe both the magnitude (size or numerical value) and the direction.

The Meaning of Force by the Physics Classroom

Object on which force is acting

The implication is that a force is just a vector because this is what it takes to “fully describe” a force. However, they contradict themselves by talking about the force “acting upon an object”. Part of the specification of a particular force is the object on which it’s acting.

This might seem like pedantry and semantics. HOWEVER, bear with me (especially as I get distracted by side issues!… and feel free to have a beer with me while you bear with me… sorry, awful pun, COVID lockdown’s been going on a while in Melbourne!)… and bear in mind that people have written endlessly about the struggles of teaching and learning Newtonian mechanics, so my claim is that this “pedantry” and “semantics” is actually “careful pedagogy”, and “care in the language we use when teaching” (which is part of careful pedagogy).


Sort of an aside, definitely slightly less important (because it’s not central to this particular post), as mentioned earlier in The Meaning of Force, for any physical quantity (including vector quantities) we need to specify the unit.

Rant: It’s frustrating that the Physics Classroom spells “newton” with a capital “N”. This may just be a minor oversight/mistake/poor editorial control… but potentially indicates the writer’s lack of breadth and/or depth of physics. Especially combined with other indicators e.g. the system of measurement (since 1960) that includes the newton, is the SI. Referring to it as the “metric” system is another potential hint of limited knowledge/understanding.

As stated in the concise summary of the SI (my bold):

Unit symbols may be more than a single letter. They are written in lower case letters, the exception being that the first letter is a capital when the unit is named after a person. However, when the name of a unit is spelled in full, it should begin with a lower case letter (except at the beginning of a sentence), to distinguish the unit from the person (for example a temperature of 293 kelvin).

A concise summary of the International System of Units, SI

Contact / non-contact

And my mind wanders on to other issues (I’ll get back to the main point shortly, I promise!)…

In addition to this, The Meaning of Force emphasises (it leads with) a distinction between contact and non-contact forces. Particularly at this level (in a resource which is introducing vectors in mechanics, Newton’s laws, energy momentum etc.) I would expect students to be starting to “categorise” the origin of forces as gravitational, electromagnetic, strong and weak e.g. friction results from electromagnetic forces between the surfaces. The beauty of physics being that a lot of seemingly complicated behaviour can be described by a relatively small number of concepts!

Sure, linking this to everyday experience/perception is always important… one should discuss how we perceive these forces… it’s just that it shouldn’t be the key categorisation at this level.

At a lower level when introducing the concept of a force, how we perceive forces is a useful first approach to stimulate students thinking about all the different sorts of forces they encounter to build up examples for their mental “toolbox” (as long as there’s still some indication that it’s an “apparent” distinction based on our perceptions, which will be resolved at a more advanced level).

Apologies… I’ve strayed off-topic… so, getting back on topic…

Object which causes force

Even specifying the body on which the force acts (and the units), they have not fully specified the force. One also needs to specify the object which is the cause of the force.

Given that The Meaning of Force begins:

A force is a push or pull upon an object resulting from the object’s interaction with another object.

The Meaning of Force by the Physics Classroom

It’s frustrating that they don’t follow through on this.

In one of the examples at the end they do, sort of. For example the table pushes upwards on the book. However, the approach does not “embed” this thinking consistently or in, for example, the notation.

They mention there is a force of gravity pulling down on the book. However, “gravity” is not an object. They need to specify that it’s the gravitational attraction of the Earth – the Earth being the other object in this interaction.


So, why is this important?… and surely all this information is there it’s just that sometimes it is “implicit”? When teaching being “implicit” can be very “dangerous”… we then wonder why, for example, those learning physics exhibit endless confusion around Newton’s 2nd and 3rd laws.

Will students suddenly make no mistakes if we do this “better”, as I’m about to describe?… no… but there’s a good chance it’ll lead to less mistakes and confusion, given that this approach naturally addresses common misunderstandings, before they arise, and makes it relatively easy to point out common mistakes.

So, how do we fully describe a force?

Forces are a model to discuss the interaction between two objects. Which, as mentioned, is how The Meaning of Force introduces them.

This is inherent in any introduction to forces from very young students where a force is simply a push or a pull (again, as introduced in The Meaning of Force) through to more advanced treatments.

So a force is one object pushing, or pulling, on another. So, to fully describe a force: we need to specify both objects, how hard or soft the push or pull is (introducing magnitude and, for quantitative work, units) – the fact that it’s a push or pull, defines the direction.

At a “more sophisticated level” we introduce the use of a vector to represent the magnitude and direction… hence “a force is a vector”. However, we need to be careful not to forget to emphasise that the vector is just part of the definition of a force.

So a force is fully defined by:

  • The object on which the force is acting.
  • The object causing the force.
  • The magnitude of the force…
  • …which, for quantitative work, requires us to choose a unit (commonly newtons).
  • The direction of the force (essentially, is it a push or a pull?).

Sure, we package the last two up as a vector… but this vector is not a full description of the force, by itself.

We can encapsulate all this information in mathematical notation / labels on diagrams. For example we can represent a force as FAonB the subscript tells us that the force (whose magnitude and direction are represented by the vector F) is exerted by the object A on the object B. For example FEarth on Book or just FEonB for the gravitational force of the Earth on a book.

This provides a relatively easy way, for example, to distinguish between Newton’s 2nd and 3rd laws. Newton’s 2nd law is about all the forces exerted on an object e.g. all those forces with subscript “on B”. Newton’s third law is about the pair of forces with subscripts “A on B” and “B on A” – and, just as importantly, if a pair of forces are not labelled “A on B” and “B on A” then they are not a Newton’s 3rd law force pair!

Now, this is nothing new. However, the number of resources and/or teachers that fail to use this approach is simply stunning. Especially “in full” / really committing to it e.g. they might state that forces are interactions etc. but then not follow through i.e. fail to use, or consistently use, related notation.

Partial credit

Even where it is used, it is not always used well. For example, in Australia, in the Victorian Certificate of Education (VCE) Physics Study Design (essentially the curriculum document… though that makes me want to start a whole other discussion!) it specifies labelling the forces by:

using the convention ‘force on A by B’ or F on A by B = –F on B by A

Victorian Certificate of Education (VCE) Physics Study Design

which even includes an implicit reference to Newton’s 3rd law and the related utility of the notation.

However, in the previous dot point (which should probably come after this one) it fails to use it’s own recommended notation, just as it fails elsewhere in the document:

model the force due to gravity, Fg, as the force of gravity…

Victorian Certificate of Education (VCE) Physics Study Design

[And it’s frustrating the document continually uses notation only referring to the magnitude of any force force, F (italic) rather than vectors F (bold roman).]

So it’s unsurprising that related textbooks are also inconsistent in using this notation, as well as failing to emphasise an understanding of what it takes to “fully specify a force”.

]Note: I prefer the more concise and, I think, easier to read FAonB rather than FonAbyB.]

Does this approach guarantee good teaching, and/or students who learn well and make no mistakes?

No. It’s hopefully obvious that there’s no “one thing”, no “magic bullet” that does this.

HOWEVER, not doing it opens up far more avenues for poor teaching (poor language etc.) and mistakes by students.

An approach that works at all levels

In addition, a useful aspect of emphasising a force as modelling an “interaction” between two objects/things/whatever is that the idea of interactions can be used from the initial introduction of pushes and pulls, to our current understanding of the universe in terms of interacting fields.

Experts vs novices etc.

We have to remember the difference here between “what physicists do”, and “how we teach physics”. The difference between “novices” and “experts”. For example, a physicist / anyone well versed in Newtonian mechanics (an “expert”) will usually not bother with the subscripts. As mentioned previously, physicists are lazy efficient.

But that’s OK for someone who’s “got it” (though sometimes some “experts” would be advised to use this notation to avoid mistakes they make). “Experts” are also more used to dealing with a variety of different notation, inconsistent notation etc. However, when a topic expert is trying to convey knowledge and understanding to someone who is a novice in the topic, if they wish to “teach well” they need to “step down” to the novice’s level.

When I learn something new in physics (or any other subject)… possibly through textbooks and/or journal papers… poor or inconsistent notation doesn’t necessarily stop me from learning, but it definitely slows me down. I might turn to another text book, or not bother reading the paper etc.

In terms of a school students or undergraduates, a lack of careful consideration over the language and/or notation by an instructor can be the difference between a student enjoying physics/doing well, or “failing”. It could be the “straw that breaks the camels back” and see a student drop out of a course/subject etc.

Even if a student naturally stops studying physics to focus on other interests, it could be the difference between them thinking of physics as a good/interesting subject and physicists as important, interesting people, or thinking physics is just a bunch of mumbo jumbo, and physicists are arrogant ******* who don’t care about trying to clearly convey what they do (especially as, in my experience, physics teachers, academics etc. tend to be generally more caring than the average… and in no way am I biased). And, apologies… I got a bit over-emotive there!


Make sure you take care of yourself. Don’t beat yourself up about any perceived lack of not doing right by your students in this time of COVID remote learning etc. oh, and:

Make sure your clear what it means to fully specify a force (or any other physical quantity, for that matter).

A force is not just a vector. A force is defined by:

  • The object on which the force is acting.
  • The object causing the force.
  • The magnitude of the force…
  • …which, for quantitative work, requires us to choose a unit (commonly newtons).
  • The direction of the force (essentially, is it a push or a pull?).

To complement this, I would recommend the notation FAonB which has various pedagogical advantages.

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