In Euclidean geometry, equipollence is a binary relation between directed line segments. A line segment AB from point A to point B has the opposite direction to line segment BA. Two parallel line segments are equipollent when they have the same length and direction.
A definitive feature of Euclidean space is the parallelogram property of vectors:
If two segments are equipollent, then they form two sides of a parallelogram:
If a given vector holds between a and b, c and d, then the vector which holds between a and c is the same as that which holds between b and d.
The concept of equipollent line segments was advanced by Giusto Bellavitis in 1835. Subsequently, the term vector was adopted for a class of equipollent line segments. Bellavitis's use of the idea of a relation to compare different but similar objects has become a common mathematical technique, particularly in the use of equivalence relations. Bellavitis used a special nota