A vector, as used in mathematics and the physical sciences, is a quantity with two parts, magnitude and direction. Vectors are used in many different contexts, including displacement/ navigation, electrical fields, relativity, audio analysis, and statistics. Today, analysis of vectors is primarily based on geometry, but before the 19th century, these quantities were relatively unknown, and when they were considered, they were explained algebraically. The development of calculus in the 17th and 18th century laid the groundwork for geometric development of vectors in the 1800s (Knott, 1978).
` Another mathematical concept that inspired vector analysis was the idea of complex numbers, which have two components, real and imaginary. When these numbers were visualized in a plane, they were equivalent to vectors. Many mathematicians tried to extend this system to three dimensions but were unable to do so (Labute, 2003). In 1843, William Rowan Hamilton invented a four-dimensional system that he called quaternions. The template for quaternions was:
q = w + ix + jy + kz, with w, x, y, and z representing real numbers. Hamilton called the first term (w) the scalar and the rest of the expression the vector, since it could be mapped on a space of three axes to produce a line. He developed an algebra of quaternions and lectured on them for the rest of his life (Crowe, 1967).
Meanwhile, a little known mathematician from Germany, Hermann Grassmann, was working on a “new geometric calculus” which would greatly broaden the vector concept so that it could be applied to any number of dimensions (Crowe, 1967). His new methods included components from matrix mechanics, tensor analysis, and linear algebra. Grassmann also published scientific papers using his methods to analyze problems in electrical theory. However, since Grassmann was not well known and his works were difficult to understand, few people knew what he had discovered.
Once scientists did begin to study his book, a rivalry between quaternions and vectors developed. Which one would be most advantageous in applications? One of Hamilton’s students, Peter Guthrie Tait, was enthusiastic about quaternions, and used them in his own work with electromagnetism (Labute, 2003). On the other hand, James Clark Maxwell, a classmate of Tait’s, wrote Treatise on Electricity and Magnetism, in which he acknowledged the usefulness of “quaternion ideas” with regard to thermo- and electrodynamics but suggested that a pure vector analysis would be more useful. Another physicist, William Kingdon Clifford, also favored vector analysis and converted the quaternion formula into two products which he termed the scalar product and the vector product. In modern vector analysis these are called the dot product and the cross product respectively (Labute, 2003).
Later in the 19th century, J. Willard Gibbs, a professor of physics at Yale University, began to use vector analysis in his studies of thermodynamics. He had studied both systems — quaternions and vectors — and viewed the latter system as being most applicable to his work (Knott, 1978). This application of Grassmann’s vector analysis became well known as a result of Gibbs’ practice of printing lecture notes for his students, who then circulated the notes to other mathematicians and physicists (Crowe, 1967). In 1901, these notes were published in book form as compiled by one of his graduate students (Wilson) and called Vector Analysis.
At the same time, another physicist who favored vectors, Oliver Heaviside, developed his own system of vector analysis for the purpose of examining electromagnetic fields. He had been inspired by Maxwell’ Treatise on Electricity and Magnetism, and went on to publish his influential 3-volume work, Electromagnetic Theory (Labute, 2003). Clearly, today’s physics, especially electrical engineering, thermodynamics, and navigation owe a great deal to the technique of vector analysis.
