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Often the standard introduction to the concept of scalars and vectors in physics is something along the lines of:

A scalar is a quantity that is completely described by a single number (it has no directional dependence). A vector is a quantity that requires both a magnitude (a number) and a direction to be specified in order to completely describe it.

This is then followed up by examples of the two, such as

mass, charge and distance are all examples of scalar quantities, whereas, velocity, displacement and force are all examples of vectors.

I feel what is often missed out is a physically intuitive explanation of why certain quantities are vector quantities whilst others are scalars. I have thought up of a few examples and I'm hoping that people can provide feedback, and/or suggest some intuitive examples/explanations.

An example of a scalar quantity is temperature. Indeed, the temperature at a point is completely specified by a single number. It is rotationally invariant, in the sense that facing northwards, or south-eastwards (or indeed any other direction) at the same point does not affect the temperature at that point. Therefore it has no directional dependence and is a scalar. Another example would be distance. A distance of $n$-meters measured from one point to another remains $n$-meters, regardless of which direction it is measured in and hence it is a scalar quantity.

An example of a vector quantity would be an objects velocity. In order to determine the motion of an object it is clearly not enough to simply provide the speed at which it is travelling. The object will travel in a particular direction, and so two objects travelling at the same speed, but in different directions will end up at completely different locations. Hence, in order to completely specify the motion of an object one must use a vector quantity - the objects velocity.

Finally, the force acting on a object is also a vector quantity, since it acts in a particular direction on the body. Whether the force acts from the top, bottom or sides of the body will have different effects on the body, hence its action is clearly directionally dependent and must be described by a vector (two forces with the same magnitude, but acting in different directions on the same object will have different effects on the object).

Hopefully what I've written is coherent. I'm hoping to convey this information to a person with a fairly minimal background knowledge in physics, so any feedback about it would be much appreciated.


Edit

To clear up ambiguity. I am not asking whether I have understood the concept correctly or not. This is more a question about How one should give an elementary introduction to vectors and scalars? I am not currently in education and so don't have a lecturer or fellow students to ask about this. I have been asked by someone to explain it to their teenage son and I wanted to hear the thoughts of others (most likely more capable than I) on how to teach this concept.

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closed as unclear what you're asking by ACuriousMind, sammy gerbil, John Rennie, David Z Jul 29 '16 at 20:10

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – David Z Jul 29 '16 at 20:10
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Think of it in the context of classical mechanics, or more intuitively your everyday environment.

For example, take a cup of coffee on a table. It has a certain mass. This is just a pure number, we don't think of it as having 'direction' - what would it mean for mass to have direction? This is just a physically intuitive idea of a scalar quantity.

Now push the coffee along the table; you're pushing it in a certain direction with a force of a certain magnitude; this is just a physically intuitive idea of a vector quantity.

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  • $\begingroup$ Thanks for your answer. This is my understanding of vectors and scalars (at least in elementary physical applications). Would the examples I gave in my original post (and the reasons why the quantities must be vectors or scalars) be acceptable? $\endgroup$ – user35305 Jul 29 '16 at 18:22
  • $\begingroup$ Your reasoning why temperature is a scalar is fine; the others are ok too. $\endgroup$ – Mozibur Ullah Jul 29 '16 at 18:30
  • $\begingroup$ Ok, thanks for taking a look. Just to clarify, is my reasoning of why velocity and force are both vectors correct? $\endgroup$ – user35305 Jul 29 '16 at 19:18
  • $\begingroup$ intuitively, yes. $\endgroup$ – Mozibur Ullah Jul 29 '16 at 19:24
  • $\begingroup$ @user35305: you're welcome. The only point I'd add, is that displacement is a vector and the magnitude of displacement is a distance, a scalar; this is analogous to velocity being a vector, and the magnitude of velocity is speed. $\endgroup$ – Mozibur Ullah Jul 29 '16 at 20:09
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When one drives a car, one generally wants to be aware of the direction in which they're moving. This information, along with the speed of the car, can be expressed as a vector.

When one wants to know how far they've gone, or how long it took to get there, direction doesn't matter. All they need is a number. This is a scalar.

This distinction is important because if you add two antiparallel (opposing) vectors together, some or all of one vector is cancelled out. However, going back to the analogy of a car, this could represent driving one direcion, and then driving back. In vector form, nothing has changed in the final state. In scalar form, they've gone some distance and it took some time.

Another reason for vectors vs. scalars is addition of forces. If you push against something and it moves, it's important to know in which direction it was pushed, and how much resistive force is contributing to making your life harder.

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  • $\begingroup$ Thanks for you answer. This was what I was thinking about the whole idea, and what I was trying to get across in my original post. Would you say that what I wrote is correct?! $\endgroup$ – user35305 Jul 29 '16 at 14:46
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Space is 3D quantity. So anything that is coupled with space lineraly is 3D quantity. I am intentionally using a beginner information without touching any relativistic phenomena.

Acceleration is 2 times coordinate derivative, hence coupled linearly. Force is acceleration times mass, hence coupled linearly. Electromagnetic field is also coupled linearly, though in a more complex way. There is no 3D vector in classical mechanics which is not coupled to coordinates.

Now, mass seems also to be coupled to coordinate. But it's 1D. Hm... From another side, in solid state physics there is effective mass which depends on direction, hence a 3D quantity again. There are more effective parameters, like refraction index, dielectric/magnetic permeability which can also become 3D quantity, if not invariant with direction! This can be explanation - quantities which have properties invariant in all directions, or under any rotation of the system are scalars.

This is attempt to get understanding, I cannot say this is the best understanding of scalar vs vector. We usually do not pay attention, because it is enough to write them and multiply by certain rules.

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