VectorsAndTheirComposition 2006-05-08 22:05:37 2006-05-08 22:29:11 Topic An Elementary Treatise on Quaternions % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \subsection{Chapter 1: Vectors and Their Composition} From \PMlinkname{An Elementary Treatise On Quaternions}{AnElementaryTreatiseOnQuaternions} by Peter Guthrie Tait. 1. FOR at least two centuries the geometrical representation of the negative and imaginary algebraic quantities, $-1$ and $\sqrt{-1}$ has been a favourite subject of speculation with mathematicians. The essence of almost all of the proposed processes consists in employing such expressions to indicate the DIRECTION, not the \emph{length}, of lines. 2. Thus it was long ago seen that if positive quantities were measured off in one direction along a fixed line, a useful and lawful convention enabled us to express negative quantities of the same kind by simply laying them off on the same line in the opposite direction. This convention is an essential part of the Cartesian method, and is constantly employed in Analytical Geometry and Applied Mathematics. 3. Wallis, towards the end of the seventeenth century, proposed to represent the impossible roots of a quadratic equation by going \emph{out of} the line on which, if real, they would have been laid off. This construction is equivalent to the consideration of $\sqrt{-1}$ as a directed unit-line perpendicular to that on which real quantities are measured. 4. In the usual notation of Analytical Geometry of two dimensions, when rectangular axes are employed, this amounts to reckoning each unit of length along $Oy$ as $+\sqrt{-1}$, and on $Oy'$ as $-\sqrt{-1}$; while on $Ox$ each unit is $+1$, and on $Ox'$ it is $-1$. T. Q. I. If we look at these four lines in circular order, i.e. in the order of positive rotation (that of the northern hemisphere of the earth about its axis, or opposite to that of the hands of a watch), they give $$1, \sqrt{-1}, -1, -\sqrt{-1}.$$ In this series each expression is derived from that which precedes it by multiplication by the factor $\sqrt{-1}$. Hence we may consider $\sqrt{-1}$ as an operator, analogous to a handle perpendicular to the plane of $xy$, whose effect on any line in that plane is to make it rotate (positively) about the origin through an angle of $90^o$. 5. In such a system, (which seems to have been first developed, in 1805, by Buee) a point in the plane of reference is defined by a single imaginary expression. Thus $a + b\sqrt{-1}$ may be considered as a single quantity, denoting the point, $P$, whose coordinates are $a$ and $b$. Or, it may be used as an expression for the line $OP$ joining that point with the origin. In the latter sense, the expression $a + b\sqrt{-1}$ implicitly contains the \emph{direction}, as well as the \emph{length}, of this line; since, as we see at once, the direction is inclined at an angle $\tan^-1b/a$ to the axis of $x$, and the length is $\sqrt{a^2 + b^2}$. Thus, say we have $$ OP = a + b\sqrt{-1};$$ the line $OP$ considered as that by which we pass from one extremity, $O$, to the other, $P$. In this sense it is called a VECTOR. Considering, in the plane, any other vector, $$OQ = a' + b'\sqrt{-1};$$ the addition of these two lines obviously gives $$OR = a + a' + (b + b')\sqrt{-1};$$ and we see that the sum is the diagonal of the parallelogram on $OP$, $OQ$. This is the law of the composition of simultaneous velocities; and it contains, of course, the law of subtraction of one directed line from another. 6. Operating on the first of these symbols by the factor $\sqrt{-1}$, it becomes $-b + a \sqrt{-1}$; and now, of course, denotes the point whose $x$ and $y$ coordinates are $-b$ and $a$; or the line joining this point with the origin. The length is still $\sqrt{a^2 + b^2}$, but the angle the line makes with the axis of x is $\tan(-a/b)$; which is evidently greater by $\pi/2$ than before the operation.