Hamilton, however, discovered that for space it is impossible to define an algebra which possesses all three of these proper ties. He tried to invent an algebra of vectors for space which should possess the second and third properties and discovered that there is no such algebra in which ij is expressible in the form x+vi-Fzj. Accordingly, he was forced to assume that the algebra was of order four instead of order three; and thus every quaternion is of the form yi+zj+wk where i, j, k denote unit vectors in three perpendicu lar directions and where the product of two vectors is of the form a+bi+cj+dk. That is, although the algebra of vectors in a plane has only two units, the algebra of vectors in space has four units. This discovery of Hamilton makes one realize that man cannot invent law; he can only discover the primordial laws of the universe.
When x, y, z, w are real numbers and i, j, k are the above unit vectors, any number of the form q=x+yi+zj+wk is called a (real) quaternion. A vector is simply a quater nion with x= o and a scalar is a quater nion with y = z =w=o. To make it con crete, we shall think of i as the unit vector toward the south, j as the unit vector to the east and k as the unit vector up. In the diagram, the plane of the page represents a vertical east-west plane viewed from the south ; the vectors j and k are in this plane and i points to the south. As for vectors in a plane, multiplication by a vector of length pis equivalent to a stretching in the ratio of p to I com pounded with a rotation in the counter-clockwise direction. Multi plication of any vector in the east-west vertical plane by i (on the left) is equivalent to a rotation in that plane through a right angle in a counter-clockwise direction as seen from the south. Similarly, multiplication by j or k is a rotation through a right angle in the direction indicated in the diagram. Since rotation through two right angles in the same plane reverses the direction, then
— I and also
From this (see fig. 6) we see that i rotates j into k and thus ij=k, whereas j rotates i into k reversed and thus ji= —k. Similarly, jk= —kj=i and ki= —ik= j. Substituting these values for the products, we see that the product of any two quaternions q=x+yi+zj+wk and Q= X +Yi+Z j+Iirk is qQ=a+bi+ cj+dk where a=xX —yY b=
yX+xY—wZ+zJV, c=zX+wY+xZ yiT', d=wX—zY+yZ+xlV. Any quater nion q and its conjugate q' =x—yi—zj wk satisfy the quadratic equation
+N(q) =o where
is the norm of q. Note that this gives a factorization of
which cannot be factored in the field of complex numbers.
From the above, it follows that the product afi of any two vectors has the form —a • 0+a X 0 where a • i3= ab cos 0, where a and b are the magnitudes of a and 0, respectively, and 0 is the angle through which a has to be rotated in order to make it extend in the same direction as 0; and a X 0 = ab(sin 0)-y where y is the unit vector perpendicular to a and j3 as indicated in the diagram (fig. 6). When we use a • 0 and «X (3 separately, we are using vector analysis (q.v., due to J. W. Gibbs). The amount of work done by a force F acting through the displacement 0 is F • 0. The area swept out by the vector a through displace ment 0 is a X0 which means that we have to regard area as a vector. Theorems in geometry can be easily proved with quaternions (vector analysis) , sometimes with great brevity; but its most important use is in mathematical physics.
due to H. Grassmann (1809-1877), is like quaternions in that it is an algebra in which some of the elements or numbers are vectors; but it is unlike quaternions in that the fundamental element is the point and the product of two elements of the same kind is an element of a different kind. Moreover, Ausdehnungslehre is not associative and applies to space of any number of dimensions.
W. R. Hamilton, Lectures on Quaternions (1853). This is the classic on the subject ; the long introduction is most interesting and suggestive to a layman. P. Kelland and P. G. Tait; Introduction to Quaternions (1873, 3rd ed. 1904). An excellent text book ; does not use calculus until the last quarter of the book ; con tains many applications and exercises. E. B. Wilson, Vector Analysis 0900, an expansion of a short pamphlet on the Elements of Vector Analysis
by J. W. Gibbs. This book pre-supposes a knowl edge of trigonometry and the elements of the calculus ; it contains many applications to geometry and physics ; the first two chapters, however, do not use calculus. J. B. Shaw, Vector Calculus (1922). A text-book on quaternions, with applications to physics ; intended for those familiar with the elements of calculus; mature; many exer cises. H. Grassmann, Die Ausdehnungslehre (Leipzig 1878-96). This is the classical work on this subject, but is highly technical. E. W. Hyde, Grassmann's Space Analysis (4th ed. 1906). A primer on Grassmann's work ; does not use calculus but requires more intellectual maturity than Gibbs-Wilson. (0. C. H.)