Vector Analysis and Quaternions

Vector Analysis and Quaternions

By “Vector Analysis” is meant a space analysis in which the vector is the fundamental idea; by “Quaternions” is meant a space-analysis in which the quaternion is the fundamental idea. They are in truth complementary parts of one whole; and in this chapter they will be treated as such, and developed so as to harmonize with one another and with the Cartesian Analysis1. The subject to be treated is the analysis of quantities in space, whether they are vector in nature, or quaternion in nature, or of a still different nature, or are of such a kind that they can be adequately represented by space quantities. Every proposition about quantities in space ought to remain true when restricted to a plane; just as propositions about quantities in a plane remain true when restricted to a straight line. Hence in the following articles the ascent to the algebra of space is made through the intermediate algebra of the plane. Arts. 2–4 treat of the more restricted analysis, while Arts. 5–10 treat of the general analysis. This space analysis is a universal Cartesian analysis, in the same manner as algebra is a universal arithmetic. By providing an explicit notation for directed quantities, it enables their general properties to be investigated independently of any particular system of coordinates, whether rectangular, cylindrical, or polar. It also has this advantage that it can express the directed quantity by a linear function of the coordinates, instead of in a roundabout way by means of a quadratic function. By a “vector” is meant a quantity which has magnitude and direction. It is graphically represented by a line whose length represents the magnitude on some convenient scale, and whose direction coincides with or represents the direction of the vector. Though a vector is represented by a line, its physical dimensions may be different from that of a line. Examples are a linear velocity which is of one dimension in length, a directed area which is of two dimensions in length, an axis which is of no dimensions in length. Subjects covered: • Addition of Coplanar Vectors • Products of Coplanar Vectors • Coaxial Quaternions • Addition of Vectors in Space • Product of Two Vectors • Product of Three Vectors • Composition of Quantities • Spherical Trigonometry • Composition of Rotations