Hamilton defined a vector as "a right line ... having not only length but also direction". Hamilton derived the word vector from the Latin vehere, to carry.
Hamilton's conceived a vector as the "difference of its two extreme points. For Hamilton, a vector was always a three-dimensional entity, having three co-ordinates relative to any given co-ordinate system, including but not limited to both polar and rectangular systems. He therefore referred to vectors as "triplets".
Hamilton defined addition of vectors in geometric terms, by placing the origin of the second vector at the end of the first. He went on to define vector subtraction.
By adding a vector to itself multiple times, he defined multiplication of a vector by an integer, then extended this to division by an integer, and multiplication (and division) of a vector by a rational number. Finally, by taking limits, he defined the result of multiplying a vector α by any scalar x as a vector β with the same direction as α if x is positive; the opposite direction to α if x is negative; and a length that is |x| times the length of α.
The quotient of two parallel or anti-parallel vectors is therefore a scalar with absolute value equal to the ratio of the lengths of the two vectors; the scalar is positive if the vectors are parallel and negative if they are anti-parallel.
Read more about this topic: Classical Hamiltonian Quaternions, Classical Elements of A Quaternion