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# Position & coordinates

Time and quantity, as well as length, area and volume, were easy to tell: ‘She will be here in 30 seconds’, ‘we’ll be 20 for dinner’ as well as ‘you must climb 5 metres’, ‘it’s a 64 square-metre apartment’ and ‘it takes 300 cubic metres of water to fill my pool’.

$$t=30\,\mathrm s\qquad n=20\qquad L=5\,\mathrm m\qquad A=64\,\mathrm{m^2}\qquad V=300\,\mathrm{m^3}$$

They are all just a number.

Position, on the other hand, is not only just a number.

The station is 300 metres that way!’ or ‘300 metres north-east’ or ‘300 metres at an angle of 45 degrees’. It seems like we here need to tell both how far and which way before it makes sense. Both distance and direction. Two pieces of information; not just one. A position apparently requires both the $300\,\mathrm m$ and the $45^\circ$.

Sometimes it even needs three numbers: ‘The balloon is 300 metres away if you turn north-east and look up under the cloud’. We here must tell how far (forwards), how much to turn the body (sideways) and how much to tilt the head (upwards). Three numbers to tell a position in the 3D sky; two numbers for the 2D ground.

Let’s invent a way to collect those two or three numbers when written, for example with brackets:1

$\left(\begin{matrix} 300\,\mathrm m \\ 45^\circ \\ \end{matrix}\right)\quad$ and $\quad\left(\begin{matrix} 300\,\mathrm m \\ 45^\circ \\ 30^\circ \\ \end{matrix}\right)$

Let’s call the numbers coordinates.2

Now, once in a while you might tell a position in another way: ‘Look 15 centimetres from the left edge and 20 from the bottom on the map’ or ‘the falcon is hovering 30 metres out, 40 metres in on the field 50 metres up’.

But these are solely distances? Are distances alone taking care of both distance and direction? Apparently, yes. Apparently, a bit outwards and a bit across is just as valid a 2D position as a bit out and then a turn, while we in 3D must go a bit up as well. So, we could just as well write:

$\left(\begin{matrix} 15\,\mathrm{cm} \\ 20\,\mathrm{cm} \\ \end{matrix}\right)\quad$ and $\quad\left(\begin{matrix} 30\,\mathrm m \\ 40\,\mathrm m \\ 50\,\mathrm m \\ \end{matrix}\right)$

These “contain” their direction, although not visibly. The conclusion seems to be that ‘position’ needs 2 numbers in 2D and 3 in 3D, regardless of the type of numbers.

When the units are in common and shared by all coordinates as in this last case, let’s invent a way to shorten it a bit by ‘taking’ the unit ‘out’. Then we only have to write the unit once:

$\left(\begin{matrix} 15\,\mathrm{cm} \\ 20\,\mathrm{cm} \\ \end{matrix}\right)=\left(\begin{matrix} 15 \\ 20 \\ \end{matrix}\right) \mathrm{cm} \quad$ and $\quad \left(\begin{matrix} 30\,\mathrm m \\ 40\,\mathrm m \\ 50\,\mathrm m \\ \end{matrix}\right) =\left(\begin{matrix} 30 \\ 40 \\ 50 \\ \end{matrix}\right) \mathrm m$

Note: Remember that

• we can think of ‘time’ as 1-dimensional, and same for other properties like ‘quantity’, ‘length’, ‘area’ and ‘volume’. All are in 1D which corresponds to them all containing just one number.
• ‘Position’ on the other hand is the defining property of space. And space is 3-dimensional. So, we can think of ‘position’, which contains three numbers, as a 3D property (and as a 2D property when we only use two numbers as above).

A property’s dimension apparently equals the number of its coordinates.3 Then position doesn’t feel that different after all – it is a “number” like any of the other properties, just a “number” in multiple dimensions (a “number” containing several numbers).

References:

1. coordinate’ (dictionary), Dictionary.com, www.dictionary.com/browse/coordinates (accessed May 7th, 2019)