In a coordinate system position is just a label for the points, so we can distinguish between different points. Then why we use a position vector to specify the position as it is just a label and nothing else. We can use colours if we want to label the points and it would be fine. For example, Paris is a label for a specific location on earth and it does not have magnitude or direction. Please explain it in as simple terms as you can do not use jargon.
In: Physics
What properties would that “label” have to be useful? Given two positions using these labels, is there a method that allows one to determine properties such as their relative distances or some kind of directional information? Given a label, how could you identify points that are “nearby”? How could you quantify that?
You are not wrong saying you can make up labels as much as you please. How would you actually make them useful?
Position is not just needed to distinquish between points. You also need to know other things like distance between these points, properties of lines drawn through some of the points etc.
Position isn’t just arbitrary labels. It needs to contain the information of *where a point is*. And that’s done most conveniently by specifying an origin point (the 0 point of the coordinate system) and then telling how far along each axis one needs to go.
Think of it like an address: you can ofc say that you live in “the green house” and you’d be fine if a) there aren’t a lot of houses b) the houses have already been painted. But otherwise it makes much more sense that you say you live in <city> <street name> <number>.
You pick a spot that is the “center”. We call this the origin. Every other spot can be described as being a certain distance (magnitude) and direction away from this center spot.
On the earth, we might choose the prime meridian and equator as centers of the earth. Whereever the prime meridian meets the equator, that spot on earth is the origin.
Paris is a certain amount of distance away from the origin. It’s also a certain direction away from the origin.
One way to describe where Paris is would be to say. It is at 48° 51′ 24″ N, 2° 21′ 6″ E and an elevation of 30 to 130 .
This is a direction from the center of the earth and a distance from it (the distance to what the French have defined as sea-level plus a few more meters.)
You can use shorthands for that like “Paris” or “Lutetia” or “City of Light” or “the capital of France” or even assign it some color or anything else. However when it come to the question of “Where is Paris?” you will have to translate these names into some sort of description of where it is in relation to yourself or a known point in the world.
You can say that Paris is 100km to the east and 50km to the south of you or simply say that it is three days travel on foot in the direction you are pointing or simply give the coordinates and those descriptions are all vectors.
When we ask where is place “x”? We expect an answer that either takes the form of a direction and a distance or some description that can be translated into one.
A step to the right is a vector because it describes a direction “right” and a magnitude “one step”.
A description like “take the third right and then go straight for three blocks” is actually two vectors, which we naturally can sum up to one.
In the real world we often use two dimensional vectors to describe points, because it is understood unless otherwise mentioned that you won’t tunnel into the earth or fly into the air but stay on ground-level. We treat the world as flat, because for the purpose of navigation it mostly is.
How far is Paris from Los Angeles? With a coordinate system, you can determine the difference between these two points by knowing what their coordinates are. You would not be able to do that by calling Paris blue and Los Angeles red.
If you sailor or pilot planning a trip from one place to another, you need to know the distance so you can fuel to the right among and calculate your ETAs. Also, knowing the coordinates tells what direction to head.
The coordinate system is specifically designed in such a way that you can calculate distances between any two positions, and this is its most useful property, not just labeling places. The coordinate system is also defined by its origin point – the point whose position is supposed to be zero, i.e. having zero values on all coordinates. So any other position is defined by a vector drawn between the origin point and that position.
Position is always defined relative to something else.
If you ask for my position and I say “I’m at 53 metres”, then that doesn’t mean anything to you. 53 metres away from what?
A position vector is just defining a set point as 0 and noting your position relative to that. When you give your longitude, you’re saying your position on the east/west axis relative to Greenwich. When you give your latitude, you’re saying your north/south position relative to the equator. So combinining them is just giving your position relative to a specific point on the globe. And for that to be useful, you have to say how far you are away from that point as well as what direction you’re moving in, which makes it a vector.
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