Explaining vector & scalar quantities to a layman

[Frank Castle]

I've been asked by someone with minimal background in physics to explain what vector and scalar quantities are and give examples. Here are my thoughts:

A scalar is a quantity that has a magnitude only, it is completely specified by a single number. Importantly, it has no directional dependence and as such is invariant under rotations and reflections of coordinate systems. An example of a scalar is temperature. Indeed, the temperature at any given point in a region does not change if one measures it whilst facing north, or whether one measures it whilst facing south, or north-east, or indeed whilst facing any other direction, it simply has a numerical value at any given point.

A vector is a quantity that has both a numerical value and a direction associated with it. Unlike a scalar, it has directional dependence. The direction of a vector can be quantified in terms of its components relative to the coordinate axes of a given coordinate system, with each component heuristically describing the amount that the vector points along each of the coordinate axes. The components of a vector are not invariant under rotations and reflections of coordinate systems, since they are relative to a particular coordinate system, therefore changing the coordinate system will change the components (intuitively, the coordinate axes of a coordinate system rotated relative to another will point in different directions than the unrotated coordinate system, and so the amount the vector points along each coordinate aside will change depending on the coordinate system). An example of a vector is the position of an object, for example the position of a car relative to a curb. One can describe its position whilst parallel to the curb facing the car or orthogonal to the curb facing away from the car, or indeed at any other angle the curb facing in any direction. Clearly the position of the car will depend on which reference frame (coordinate system) one measures it from. If one measures it whilst parallel to the curb facing the car then its position is parallel to a single axis of your coordinate system, whereas if one measures the position of the car at a 45 degree angle to the curb (facing the car for simplicity), then its position will have components along two axes of your coordinate system. Hence, the coordinate description of the cars position is not invariant under rotations and it is this a vector, having both magnitude (the distance from the observer to the car) and direction (the angle of car relative to the observer).

I appreciate that this isn't a question as such, but any feedback, suggested improvements of the explanation, would be much appreciated.

 

[Grinkle]

I think you can make it simpler. Wind speed is a scalar, wind speed plus wind direction is a vector.

 

[PeroK]

A scalar is described by a magnitude (numerical value) alone. For example, mass, temperature, energy, electric charge.

A vector is described by both a magnitude and a direction. For example, displacement, velocity, electric current.

 

[Frank Castle]

I think you can make it simpler. Wind speed is a scalar, wind speed plus wind direction is a vector.

Good point. I realized in hindsight that what I put was a bit convoluted (unfortunately I haven't had an opportunity until now to return to it). I think it ended up being a little long-winded because I was trying to emphasise the fact that ones coordinate description of a vector changes depending on ones orientation. Using the wind example, if one is facing towards the direction of the wind, then its vector is pointing in the same direction to oneself, however, if one rotates by 90 degrees to face away from the direction of the wind then its vector points in the same direction to oneself.

A scalar is described by a magnitude (numerical value) alone. For example, mass, temperature, energy, electric charge.

A vector is described by both a magnitude and a direction. For example, displacement, velocity, electric current.

How would you argue that particular quantities are either vectors of scalars though? This is what I was attempting to do alongside my explanation. For example, why temperature is a scalar quantity whereas velocity is a vector quantity. I was trying to do so by explaining the invariance of a scalar under rotations of coordinate systems, and then for vectors, the fact that they have a direction associated with them means that ones coordinate description of them changes depending on how one is oriented with respect to the vector quantity.

I guess I'm also trying to motivate why we need such quantities. My argument would be that certain quantities have intrinsic direction associated with them whereas others do not. For example, one exerts a force in a particular direction with a given magnitude (the angle that one exerts a force on an object changes the amount of force acting on that object, hence it is a vector quantity), another example would be a car traveling in a particular direction at a given speed, this is quantified by velocity - if the car changes direction, then its velocity changes.
Other quantities do not have any direction associated with them, for example temperature, which has a single value at each point, and no direction associated with it (ones orientation at a given point doesn't affect the temperature one measures at that point), or the mass of an object (again, the measured value doesn't depend on how one is oriented with respect to the object).

 

[PeroK]

How would you argue that particular quantities are either vectors of scalars though? This is what I was attempting to do alongside my explanation. For example, why temperature is a scalar quantity whereas velocity is a vector quantity. I was trying to do so by explaining the invariance of a scalar under rotations of coordinate systems, and then for vectors, the fact that they have a direction associated with them means that ones coordinate description of them changes depending on how one is oriented with respect to the vector quantity.

Inavariance under changes of coordinate systems is a much deeper and more sophisticated view of scalars and vectors than is needed. For a basic understanding you don't even need to introduce coordinate systems.

Who's this designed for? You said with a "minimal background in physics". Well, for someone like that you're not going to elucidate anything by taking about invariance under rotations of coordinate systems.

 

[Stephen Tashi]

A vector is a quantity that has both a numerical value and a direction associated with it.

That's a typical description of the concept of "vector" in applications to physics, but there can be vectors of test scores, vectors of stock prices and so forth in other disciplines. In physics there are infinite dimensional vector spaces. I don't know what concept "direction" conveys in such a context.

We can probably find examples of things that have magnitude, "direction" and a notion of "addition", but aren't vectors because they don't obey the parallelogram law when added. Try traffic flows on intersecting streets.

I suppose simple explanations are only approximations.

 

[Frank Castle]

Inavariance under changes of coordinate systems is a much deeper and more sophisticated view of scalars and vectors than is needed. For a basic understanding you don't even need to introduce coordinate systems.

Who's this designed for? You said with a "minimal background in physics". Well, for someone like that you're not going to elucidate anything by taking about invariance under rotations of coordinate systems.

I guess you're right. As far as I know, the person has a high school standard knowledge of maths and physics, but finished school a long time ago, so their knowledge is very rusty.

Perhaps it would be better to say something like:
Certain quantities can be fully described by a single number, such as the temperature at a given point, or the mass of an object, etc. However, some quantities cannot be fully described by a single number, they also have a direction associated with them. An example would be the motion of a car; to completely describe its motion, one needs to specify the speed it is traveling at and also in what direction it is traveling in (simply providing a speed for the car does not provide enough information to fully determine its motion - traveling at 20mph north-east or 20mph south-west will result in the car ending up in a completely different location). Another example would be force; to fully describe a force acting on a body one needs to specify how hard the force pushes or pulls on the body and also in what direction this push or pull is acting on the body (if a direction is not specified then we do not know whether the body is pushed/pulled from left to right, up to down, etc. - the action of the force on the body is clearly direction dependent).

The reason why I'm trying to emphasise the importance of the direction associated with a vector quantity is because the person is confused by why one needs vectors to describe things in the first place. I'm trying to rationalise with them that the reason is we want to be able to determine things such as the motion of an object, its position and how it changes. My argument was that, if we didn't specify a direction then we wouldn't know where the object ends up after we have moved it - simply stating that it has moved 5 meters (for example) does not give us enough information to specify where it is now located. Furthermore, other quantities, such as force, also intrinsically have a direction associated with them. A force acts on an object in a particular direction - in other words the push or pull exerted by the force occurs along a particular direction. Simply stating the magnitude of the push or the pull does not provide enough information, since a push or pull with the same magnitude, but in two different directions, has different effects on an object. Thus, to fully describe the effect of a force one needs to specify its magnitude and the direction in which it it is acting, and so it is a vector quantity.

That's a typical description of the concept of "vector" in applications to physics, but there can be vectors of test scores, vectors of stock prices and so forth in other disciplines. In physics there are infinite dimensional vector spaces. I don't know what concept "direction" conveys in such a context.

We can probably find examples of things that have magnitude, "direction" and a notion of "addition", but aren't vectors because they don't obey the parallelogram law when added. Try traffic flows on intersecting streets.

I suppose simple explanations are only approximations.

Yes, I realize what I've presented is quite a simplistic view, but I'm trying to explain it to some from a basic physics perspective.

 

[David Lewis]

An example of a vector is the position of an object, for example the position of a car relative to a curb. Clearly the position of the car will depend on which reference frame (coordinate system) one measures it from.

Only the position vector will depend on your choice of reference frame, not the position of the car.

 

[Frank Castle]

Only the position vector will depend on your choice of reference frame, not the position of the car.

Yes, that's true. Apologies, my original explanation was a little convoluted. Perhaps a more succinct explanation for why position is a vector quantity is found from noting that the position of an object relative to a given reference point can clearly not be described by just its distance from that point alone. Indeed, two objects can be equidistant from the same point, but located at completely separate points. As such, in order to specify the location of an object, relative to a given reference point, i.e. its position relative to this point, one clearly needs to specify both the distance and the direction of the object relative to this reference point. Hence, position is a vector quantity.

 

[lychette]

your reference to a car journey in #7 is a good place to start in my experience.
Drive 3 miles east then drive 4 miles north...how far have you traveled (distance) ..7 miles
How far away from the start (displacement) ..5 miles.
Displacement is a vector, distance is a scalar. strictly speraling the displacement is 5 miles at a bearing of about 36⁰
School kids recognise this difference in terms. a 400m race has a distance of 400m but a displacement of 0.
hope this helps

 

[OldbutnotDead]

I like a bowling description. The bowling ball has a set weight. When you choose the ball you automatically start doing simple vectors in your head. You do a test swing imagining how hard to slide, how to angle your body, your point of reference lining up the ball. It can be described with numbers but you are already using the principals without thinking about it. Btw I'm like your friend not much education yet but will be changing that soon.

 

[anton_kiziloff]

In my experience the simplest questions are the hardest to answer. Because they require to press all the physical and mathematical background in one sentence. :) Perhaps you've already answered your question yourself, mentioning the necessity for a physical quantity to have a direction and an orientation in space. But still, I would recommend to tell your friend that everything in physics is about a motion in space. And space is not just a railroad, where you can move either forward or backward. It has at least 3 dimensions, adding 2 more components in a description of a physical quantity. Of course the are such cases as Work, or Energy. But with Work you can argue that this is a measure of Force influence on motion of a body. As for Energy... Good luck with that!