Quaternions as the result of algebraic operations . e designated by i, ], k. 25. So far we have recognized two kinds of magnitudes,scalars and vectors, and six operations, addition, subtrac-tion, reversion, multiplication, division, and mean reversion. Applying these operations to scalars we find that theyall produce scalars again, except in the case of mean rever-sion, and that produces a vector. This gave us the secondkind of magnitude, to which we will now proceed to applythe six fundamental operations. 26. Reversal is merely the turning of the vector into theopposite direction, as the word
Quaternions as the result of algebraic operations . e designated by i, ], k. 25. So far we have recognized two kinds of magnitudes,scalars and vectors, and six operations, addition, subtrac-tion, reversion, multiplication, division, and mean reversion. Applying these operations to scalars we find that theyall produce scalars again, except in the case of mean rever-sion, and that produces a vector. This gave us the secondkind of magnitude, to which we will now proceed to applythe six fundamental operations. 26. Reversal is merely the turning of the vector into theopposite direction, as the word implies. The result is some-times called a revector. 27. Addition and subtraction of vectors. Subtraction ofvectors is merely addition with the minuend are of the nature of strokes, with the property ofabsolute direction added. The laws governing the addition of strokes evidently holdhere also. Thus, vector ad-dition is commutative andassociative, and this whetherthe vectors are coplanar ornot. Thus a+fi + y=(i+f+a = r+a+/?,etc.,. SPACE IDIOGRAPHS 13 where a, /?, y are the three edges of a parallelopiped. Thesame reasoning would apply to additional vectors. 28. The following equations are self evident: a +a +« ... to m terms =wia,— rt + ( —«) + ... to w ( —a) = —ma,ma+mft = ni{a+fi), ma — m^^mia—^). 29. If «, /?, ;- be three coinitial vectors, then any fourthcoinitial vector, f, can be expressed as ^=xa-\-y[i+ZY, f being the diagonal of the parallelopiped whose edges arexa, !//3, and zy. 30. If a is a unit vector, and ma=A, then m indicatedgenerally by the symbol TA, which expresses the length ofthe vector A, is called the tensor {tendere, to stretch) of thevector A. a, denoted by UA is called the unit vector ofA. Therefore A=TA-VA. Vectors will be denoted by capital Greek letters when thetensor and unit part are to be emphasized; by lower-caseGreek letters when the question of length is not important;and by the corresponding lower-cas
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