Introduction to Quaternions

1. Definition of a Vector. A vector is the representative of transference through a given distance, in a given direction. Thus if AB be a straight line, the idea to be attached to 'vector AB' is that of transference from A to B.

For the sake of definiteness we shall frequently abbreviate the phrase 'vector AB' by a Greek letter, retaining in the meantime (with, one exception to be noted in the next chapter) the English letters to denote ordinary numerical quantities.

If we now start from B and advance to C in the same direction, BC being equal to AB, we may, as in ordinary geometry, designate 'vector BC' by the same symbol, which we adopted to designate 'vector AB.'

Further, if we start 'from any other point O in space, and advance from that point by the distance OX equal to and in the same direction as AB, we are at liberty to designate 'vector OX' by the same symbol as that which represents AB.

Other circumstances will determine the starting point, and individualize the line to which a specific vector corresponds. Our definition is therefore subject to the following condition: -- All lines which are equal and drawn in the same direction are represented by the same vector symbol.

We have purposely employed the phrase 'drawn in the same direction' instead of 'parallel' because we wish to guard the student against confounding 'vector AB' with 'vector BA.'