Last time we discussed reference frames and coordinate systems which may have been a little too abstract so let’s put it into perspective with an example inspired by Taylor and Wheeler’s Parable of the Surveyors from Spacetime Physics (let me know if you would like an ebook version – it’s a great introduction to relativity).
Crusoe wishes to map the island he is stranded upon. He doesn’t have a compass but he takes a guess at which direction is north and calls it “northish”. Being fond of the Cartesian coordinate system, Crusoe records the location of various landmarks by how many meters northish and eastish he must walk to reach them. Here is a sample of his findings:
| Landmark | Distance northish | Distance eastish |
|---|---|---|
| Cave | 0 | 0 |
| Waterfall | 0 | 15 |
| Kissing rocks | 20 | -75 |
| Highest peak | 100 | 60 |
| Ruins | 70 | 40 |
Coordinates such as these can be put together to form a mathematical object called a vector. Vectors are commonly represented as 
Vectors can be added and subtracted like numbers. To add vectors, one simply adds the corresponding components. The result is a new vector that is equivalent to where you would be if you were to walk to the coordinates of the first vector and then pretended you were back at the cave and walk the distance and direction given by the second vector. For example 
To find the length of a vector (ie the distance between the cave and the location represented by the vector) one applies Pythagoras’ theorem and adds the squares of the northish component and the eastish component and takes the square root of this result. Thus the distance to the ruins is 
One unfortunate day, pirates arrive on Crusoe’s island on the hunt for buried treasure. The pirates have a map that has been constructed using a compass, so the coordinates are given in distances north and east and not Crusoe’s invented directions northish and eastish:
| Landmark | Distance north | Distance east |
|---|---|---|
| Cave | 0 | 0 |
| Waterfall | 12 | 9 |
| Kissing rocks | -48 | -61 |
| Highest peak | 108 | -44 |
We say the pirates’ map has been constructed with a different coordinate system. Because each coordinate vector corresponds to a real location on the island, each vector from the pirates’ map corresponds to a vector in Crusoe’s system. The notation 
Note that the length of a vector should be the same whether it is measured in Crusoe or pirate coordinate system (provided they used the same units). Mathematically we say 
When Crusoe approaches the pirates, their immediate instinct is to kill him, but being (pi)rational individuals they realise that his local knowledge may help them locate their treasure. The map describes a location that Crusoe recognises as some ancient ruins he happened upon, but he only knows their location as 70 m northish and 40 m eastish or 
