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SECTION 4.-On Coefficients of Vectors.

12. The simple or single vector, a, is also denoted by la, or by 1. a, or by (+1) a; and in like manner, the double vector, a+a, is denoted by 2a, or 2. a, or (+ 2) a, &c. ; the rule being, that for any algebraical integer, m, regarded as a coefficient by which the vector a is multiplied, we have always,

la + ma = (1 + m) a ;

the symbol 1 + m being here interpreted as in algebra. Thus, Oa = 0, the zero on the one side denoting a null coefficient, and the zero on the other side denoting a null vector; because by the rule,

la + Oa = (1 + 0) a = la = a, and .'. Oa = a - a = 0.

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Again, because (1) a + (− 1) a = (1 − 1 ) a = 0a = 0, we have (-1) a = 0 -a=-a=-(la); in like manner, since (1) a + (− 2) a = (1-2) a = (-1) a = -a, we infer that (-2) a = -a - a = − (2a); and generally, (m) a = - (ma), whatever whole number m may be so that we may, without danger of confusion, omit the parentheses in these last symbols, and write simply, - la, - 2a,

- ma.

13. It follows that whatever two whole numbers (positive or negative, or null) may be represented by m and n, and what



Fig. 12.


ever two vectors may be denoted by a and ß, we have always,

as in algebra, the formula,

na + ma = (n + m) a, n (ma) = (nm) a = nma,

and (compare Fig. 12),

m (B±a) = mB + ma ;



so that the multiplication of vectors by coefficients is a doubly distributive operation, at least if the multipliers be whole numbers; a restriction which, however, will soon be removed.

14. If ma = ẞ, the coefficient m being still whole, the vector B is said to be a multiple of a; and conversely (at least if the integer m be different from zero), the vector a is said to be a sub-multiple of ß. A multiple of a sub-multiple of a vector is said to be a fraction of that vector; thus, if ß=ma, and y = na, then y is a fraction of ẞ, which is denoted as follows, y


= = ß;



also ẞ is said to be multiplied by the fractional coefficient m andy is said to be the product of this multiplication. It follows that if x and y be any two fractions (positive or negative or null, whole numbers being included), and if a and ß be any two vectors, then

ya ± xa = (y + x)a, `y(xa) = (yx) a = yxa, x(ẞ±a) = xẞ ± xa ;

results which include those of Art. 13, and may be extended to the case where x and y are incommensurable coefficients, considered as limits of fractional ones.

15. For any actual vector a, and for any coefficient x, of any of the foregoing kinds, the product xa, interpreted as above, represents always a vector ß, which has the same direction as the multiplicand-line a, if x > 0, but has the opposite direction if x < 0, becoming null if x = 0. Conversely, if a and ẞ be any two actual vectors, with directions either similar or opposite, in each of which two cases we shall say that they are parallel vectors, and shall write ẞ a (because both are then parallel, in the usual sense of the word, to one common line), we can always find, or conceive as found, a coefficient x≥0, which shall satisfy the equation ẞ = xa; or, as we shall also write it, Bax; and the positive or negative number x, so found, will bear to +1 the same ratio, as that which the length of the line B bears to the length of a.


16. Hence it is natural to say that this coefficient x is the quotient which results, from the division of the vector B, by the parallel vector a; and to write, accordingly,

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so that we shall have, identically, as in algebra, at least if the divisor-line a be an actual vector, and if the dividend-line ẞ be parallel thereto, the equations,

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which will afterwards be extended, by definition, to the case of non-parallel vectors. We may write also, under the same


conditions, à 2, and may say that the vector a is the quotient


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of the division of the other vector ẞ by the number x; so that we shall have these other identities,

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17. The positive or negative quotient, x = B, which is thus


obtained by the division of one of two parallel vectors by another, including zero as a limit, may also be called a SCALAR; because it can always be found, and in a certain sense constructed, by the comparison of positions upon one common scale (or axis); or can be put under the form,

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where the three points, A, B, C, are collinear (as in the figure annexed). Such scalars are, therefore, simply the REALS (or real quantities) of Algebra; but, in combina



Fig. 13.

tion with the not less real VECTORS above considered, they form one of the main elements of the System, or Calculus, to



which the present work relates. In fact it will be shown, at a later stage, that there is an important sense in which we can conceive a scalar to be added to a vector; and that the sum so obtained, or the combination,


"Scalar plus Vector,"



SECTION 1.-On Linear Equations connecting two Co-initial Vectors.

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18. WHEN several vectors, OA, OB, . are all drawn from one common point o, that point is said to be the Origin of the System; and each particular vector, such as oa, is said to be the vector of its own term, a. In the present and future sections we shall always suppose, if the contrary be not expressed, that all the vectors a, ß, . . which we may have occasion to consider, are thus drawn from one common origin. But if it be desired to change that origin o, without changing the termpoints A,.. we shall only have to subtract, from each of their old vectors a, . . one common vector w, namely, the old vector oo' of the new origin o'; since the remainders, a w, B-w,.. will be the new vectors a', ẞ',.. of the old points A, B, For example, we shall have


a' = O'A = A − 0′ = (1 − 0) − (0′ − 0) = 0A – 00′ = a - w.

19. If two vectors a, ẞ, or oA, OB, be thus drawn from a

given origin o, and if their

directions be either similar or

opposite, so that the three

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points, o, A, B, are situated on one right line (as in the figure


annexed), then (by 16, 17) their quotient is some positive or


negative scalar, such as x; and conversely, the equation B = xa, interpreted with this reference to an origin, expresses the condition of collinearity, of the points o, A, B; the particular values, x 0, x = 1, corresponding to the particular positions, o and A, of the variable point в, whereof the indefinite right line oa is the locus.


20. The linear equation, connecting the two vectors a and B, acquires a more symmetric form, when we write it thus:

aa + bB = 0;

where a and b are two scalars, of which however only the ratio is important. The condition of coincidence, of the two points

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since we do not suppose that both the coefficients vanish; and the equation ẞ= a, or OB = OA, requires that the point в should coincide with the point A: a case which may also be conveniently expressed by the formula,

B = A;

coincident points being thus treated (in notation at least) as equal. In general, the linear equation gives,

a. OA + b. Oв= 0, and therefore

a: b

= BO: OA.

SECTION 2.-On Linear Equations between three co-initial


21. If two (actual and co-initial) vectors, a, ß, be not connected by any equation of the form aa + bẞ= 0, with any two scalar coefficients a and b whatever, their directions can neither be similar nor opposite to each other; they therefore determine

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