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which express the associative and commutative laws for addition and subtraction.

It is evident that the principles of geometric addition apply equally to vectors in space, or in the plane, or to segments of one straight line. In particular^ algebraic addition (Art. 3) may be described as geometric addition in a straight line.

60. Geometric Multiplication. The geometric product of two magnitudes ^a, /?, is denned as a third magnitude y, whose tensor isjthe algebraic product of the tensors of the factors and whose amplitude is the algebraic sum of their amplitudes, constructed by the rules for algebraic product and sum. (Arts. 5, 3.)

If one of the factors be real and positive, the amplitude of the other reappears unchanged as the amplitude of the product, which is then constructed, by the algebraic rule, upon' the straight line that represents the direction of the complex factor, and it was proved in Art. 18 that in such a construction an interchange of factors does not change the result. Hence, if a be real and posit1ve and /? complex,

a X- £.= ^ x a- ^ In this product, tensor of a X (5 = a X tensor of ft by definition, and if tensor of /S=1, the complex quantity a X /? appears as the product of its tensor and a unit factor /3, a complex unit, which when applied as a multiplier to a real quantity a, does not change its length but turns it out of the real axis into the direction of /?. fy^s Any such complex unit is called a versor. Cty^ Let tsr stand for tensor, vsr for versor; y/C then every complex quantity a can be 2^ expressed in the form

FiX- 3'- a = tsr a X VSf a.

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This versor factor is wholly determined by its amplitude, in terms of which it is frequently useful to express it. For this purpose let i be the versor whose amplitude is Tt/2, 6 the amplitude of the complex unit /3, OX the real axis, BM the perpendicular to OX from the terminal extremity of /?. Then MB' = sin 6, OM= cos 6,

and by the rule of geometric Q l M A ^ addition. Fig. 3*

P = OM-+ i X MB = cos 6 + z sin 0. As an abbreviation for cos 6 -f- z sin 0 it is convenient to use cis 6, which may be read: sector of 6. In this symbolism, the law of geometric multiplication (product of complex quantities, as above defined) is expressed in the formula, ". (a cis <£) X (J> ' cis rf) = a X b ' cis (<£ -f

It is obvious that algebraic multiplication, described in Art. 5, is a particular form of geometric multiplication, being geometric multiplication in a straight line.

61. Conjugate and Reciprocal. If in the last equation b = a and ij<= — <£, it becomes

(a cis 40 X (a ' cis [— <£]) = a X o. • cis o = a2.

The factors of this product, a cis <f> and a cis (— are said to be conjugate to one another, and we have the rule:

The product of two conjugate complex quantities is equal to the square of their tensor.

If this tensor be 1, the product reduces to cis 4> ' cis (— 4>) = 1;

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The letters here involved may .obviously represent either direct factors or reciprocals, and the sign X may be replaced at pleasure by the sign /, without affecting the proof here given (see Art. 61). Hence, as in Arts. 32, 33, with real magnitudes, so with complex quantities the associative and commutative laws for multiplication and division have their full expression in the formulae,

y/ Q9«y/ /?)=>9 (* «) 09 /»),

65. The Distributive Law. From the definitions of geometric addition and multiplication (Arts. 58, 60) the law of distribution for complex quantities is an easy consequence. The constructions of the subjoined figure, in which a, /J and m ' cis 6 represent complex quantities will g bring this law in direct evidence. The operation wcis6 changes OA into OA', AB= A'F into AE= A^B'', and OB into OB', that is, turns each side of the triangle OAB through the angle AO A' and changes its length in the ratio of wto 1, producing the similar triangle OA'B', in which

OA' = m- cis 6 X a,
A'B' = m . cis 6 X A
OB' = m . cis6 X (a + /?).

But, by the rule of geometric addition,

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