Skill 11 of 13
In Progress

Position & coordinates

Technical skill ●●●○○

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 single number.

Position, on the other hand, is not 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 say both how far and which way before it makes sense. Both distance and direction. Two pieces of information, not just one. A position apparently needs 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.’ Here we must say 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.

We might want to group those two or three numbers when written since they only make sense together. Why don’t we wrap them in brackets like this:1

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

Let us call the numbers within the brackets coordinates.2

Once in a while you might talk about positions 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 in, 40 metres out on the field and 50 metres up.

But are these coordinates 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 (and in 3D we must go a bit up as well). So we could just as well write the positions as:

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

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

When the units are the same for all coordinates, let us 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 \text{ 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 

Remember that

  • we can think of time and also quantity, length, area and volume as one-dimensional. All are in 1D, which corresponds to them all containing just one number (only one “coordinate”).
  • Position on the other hand is the defining property of space, and space is three-dimensional. So we can think of position, which contains three numbers (coordinates), 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)