QUATERNIONS is the algebra of vectors—quantities, like force, velocity and acceleration, that have both magnitude and direction. Ordinary numbers (scalars) have merely magnitude.
All positive numbers (integral, fractional or irrational) may be represented by the points of a straight line lying on one side of some point 0 which we conceive as corresponding to zero, as if the straight line were the indicator of a thermometer (fig. I). If the line be horizontal, then the positive numbers are usually rep resented by the points to the right of 0, the zero-point, and the negative numbers correspond to the points to the left of 0 (tem peratures below zero). The totality of numbers corresponding to the points on this line are called real num bers. Thus far, the numbers introduced represent magnitude only and there is no di rection implied by a number except the dis tinction between backward and forward. Vectors in a Plane.—If we wish to have numbers which indicate direction in a plane as well as distance we can use ordinary complex numbers (see COMPLEX NUMBERS). To do this, we note that, since ( — I) a= —a, then multiplication of any number by negative one rotates the line segment OA representing a through two right angles. But, since —1 V —1= — I, this shows that multiplication by V — I must rotate OA through one right angle.
Thus we can represent any point in the plane of this page by a number x+yi where x and y are real numbers and where i is written for V — I. Better still, (fig. 2) we can think of the number x-Fyi as repre senting the directed line segment OQ, or vector as we say (see VECTOR ANALY sis).
One of the most familiar examples of a _ vector is position of a point with respect to some other point, e.g., the position of a building is four blocks south and two blocks east of the city hall. Other examples are displacement through a certain distance in a certain direction, velocity, acceleration, momentum, force, rota tion. Vectors are contrasted with ordinary numbers or scalars, examples of which are distance, time, temperature, volume, mass, work, energy (figs. 3 and 4).
Since the displacement AB followed by the displacement BC has the same effect as the single displacement AC, then the sum of the two vectors AB and BC is the vector AC. The multipli cation of any number a-Fbi by the number x+yi means per forming on the vector OP=a+bi the same operation which changes the vector OU=I into the vector OQ =x-Fyi, and hence is a rotation of the vector OP through the angle 0, followed by a stretching in the ratio of p to 1 where p is the length of OQ (shrinking,if p be less than 1) and thus obtain the vector OR. of ordinary complex numbers obeys the commuta tive law—that is, if u and v be any two complex numbers, then uv = vu; and it also obeys the as sociative law—that is, if a, v and w be any three complex numbers, then (uv)w=u(vw). (See the articles COMMUTATIVE and ASSO CIATIVE LAWS, respectively.) These two laws are so familiar that we have taken them for granted, but we shall have to dis card one of them when we come to quaternions. Thus the field of ordinary complex numbers is the algebra of distances and direc tions in a plane. (See LINEAR ALGEBRAS.) Vectors in Space.—When Sir W. R. Hamilton (1805-1865) tried to invent an algebra for vectors in space, he discovered some peculiarities which at first seemed disconcerting. By analogy with the algebra of vectors in a plane, he tried to represent every vector in space in the form x+yi+zj where I, i and j denote unit vectors in three mutually perpendicular directions (fig. 5a). This seems natural, inasmuch as any vector (displacement, force, etc.) in space is compounded of a vector in a north-and-south direction, a vector in east-west direction and a vector in an up-and-down direction. Now in the field of complex numbers multiplication obeys, (I) the commutative law, and (2) the associative law, while (3) division— the inverse of multiplication—is uniquely possible when the divisor is not zero.