A man comes out of a small hut, and walks a mile due south. He then walks another mile due east, at which point he sees a bear. The hut he started from is now exactly a mile away. What colour was the bear?

Situation (i) Two aeroplanes are flying in the neighbourhood of an airport on defined paths. Will they collide?

Situation (ii) A satellite is on a trajectory towards Jupiter. Measurements made near the end of its journey indicate that the relative positions of Jupiter and its moons are not quite as expected. How far off course is the satellite?

Situation (iii) A telescope is designed so as to be able to investigate a particular star over a period of several months. Given that the earth rotates on its axis, and also revolves around the sun, how does the required orientation of the telescope vary with time?

Situation (iv) An experiment requires a large and complicated apparatus, the various components of which are being constructed in several different laboratories. When it is to be finally assembled, it is crucial that the separate parts which are supposed to fit together do so; and also that the different components and their associated electrical leads, vacuum pipework, etc. do not lay claim to the same region of space. […]