Grammaticality – Understanding ‘Either of Them’ in a Sentence

Welcome to EL&U, Sukessh. In most circumstances 'either' means 'any one of two'. For instance "You can have either the pink ball or the blue ball" means that you can have the pink ball or you can have the blue ball but you cannot have both.

This element of binary choice exists even if there are more than two objects to choose from so long as each one falls exclusively into one of two classes. For instance, with two boxes of fruit available a parent might tell a child "You can have either an apple or an orange", the child can choose one fruit from one of the classes even though there are many of each available.

The element of binary choice exists in other contexts too, "You can take either the left or the right turn, they both go the Market Place" means that, if your destination is the Market Place, the two routes are equivalent.

There are cases where the choice is between two supposedly identical things. When gentlemen were still stupid enough to have duels over points of honour they might be offered a choice from a pair of duelling pistols from which they had to take one. The two pistols were as near identical in form as the technology of the time could make them, they were loaded with identical powder and balls and the first combatant could "take either of them".

Also if you are playing backgammon using two dice and your last two pieces are on the first and sixth points from home you can get out if either of the dice you throw next time is a six because the other piece can be borne off whatever the other dice shows. In this case a double six still fits the condition because one of the dice shows a six. The fact that they both fit the condition is irrelevant.

When it comes to vectors the term 'either' still expresses a binary choice but the conditions are different. If I say that there is a case where the resultant vector AB can be equal to either vector A or vector B depending on how A and B are defined I am saying applying AB can take us to the same place on the cartesian plane as applying either A or B and that the sizes of the resultant and one of the contributing vectors are the same. Looking at the diagram in the Quora post it is obviously not true that AB takes us to where B takes us because both B and AB start at the same point but it does take us to where A takes us and AB and A have the same size as they are sides of an equilateral triangle. What the 'either' means in this case is that, even if the vectors had been labelled differently, the resultant would be equal to one of the contributing ones.

In the case of zero length vectors we can still say that the resultant is equal to 'either' of the contributing ones since there is a contributing vector which takes us to the same place and has the same length. The fact that both the contributing vectors are identical to each other and to the resultant doesn't matter, the choice is analogous to the choice between dice showing a six, duelling pistols, routes to the market place or between one apple (or orange) and another. In this case 'either' still does not mean 'both', it's just that the choice is arbitrary and the case mathematically trivial.