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3. An equation between vectors, considered as an equidifference of points, admits of inversion and



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SECTION 2.-On Differences and Sums of Vectors taken two by two.

4. In order to be able to write, as in algebra,

(c' - a) − (b − a) = C – B, if c' – A' = c — A,

we next define, that when a first vector AB is subtracted from a second vector ac which is co-initial with it, or from a third vector a'c' which is equal to that second vector, the remainder is that fourth vector BC, which is drawn from the term B of the first to the term c of the second vector: so that if a vector be subtracted from a transvector (Art. 1), the remainder is the provector corresponding. It is evident that this geometrical subtraction of vectors answers to a decomposition of vections (or of motions); and that, by such a decomposition of a null vection into two opposite vections, we have the formula,

0-(BA) = (AA) - (B - A) = A - B ;

so that, if an actual vector AB be subtracted from a null vector AA, the remainder is the revector BA. If then we agree to abridge, generally, an expression of the form 0- a to the shorter form, -a, we may write briefly, - AB = BA; a and being thus symbols of opposite vectors, while a and − (− a) are,


for the same reason, symbols of one common vector: so that we may write, as in algebra, the identity,

-(-a) = a.

5. Aiming still at agreement with algebra, and adopting on that account the formula of relation between the two signs, + and

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in which we shall say as usual that b - a is added to a, and that their sum is b, while relatively to it they may be jointly called summands, we shall have the two following consequences :

I. If a vector, AB or B-A, be added to its own origin a, the sum is its term в (Art. 1); and

II. If a provector BC be added to a vector AB, the sum is the transvector Ac; or in symbols,

I. . (B − A) + A = B; and II.. (c − B ) + (B − A) = C − A.

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In fact, the first equation is an immediate consequence of the general formula which, as above, connects the signs + and -, when combined with the conception (Art. 1) of a vector as a difference of two points; and the second is a result of the same formula, combined with the definition of the geometrical subtraction of one such vector from another, which was assigned in Art. 4, and according to which we have (as in algebra) for any three points, A, B, C, the identity,

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It is clear that this geometrical addition of successive vectors corresponds (comp. Art. 4) to a composition of successive vections, or motions; and that the sum of

two opposite vectors (or of vector and revector) is a null line; so that

BA + AB = 0, or (A - B) + (B − a ) = 0. It follows also that the sums of equal pairs of successive vectors are equal; or more fully that


Fig. 7.

if B'A' BA, and c'-B'C-B, then C-AC-A;




the two triangles, ABC and A'B'C', being in general the two opposite faces of a prism (comp. Art. 3).

6. Again, in order to have, as in algebra,

(c' — B') + (B — A) = c - A, if c' - B' = C - B,

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we shall define that if there be two successive vectors, AB, BC, and if a third vector B'c' be equal to the second, but not successive to the first, the sum obtained by adding the third to the first is that fourth vector, AC, which is drawn from the origin A of the first to the term c of the second. It follows that the sum of any two co-initial sides, AB, AC, of any parallelogram ABDC, is the intermediate and co-initial diagonal AD; or, in symbols,

(C−A) + (B − A) = D - A, if D-C = B− A;


because we have then (by 3) c— A =D - B.


Fig. 8.



7. The sum of any two given vectors has thus a value which is independent of their order; or, in symbols, a + B = ß + a. If equal vectors be added to equal vectors, the sums are equal vectors, even if the summands be not given as successive (comp. 5); and if a null vector be added to an actual vector, the sum is that actual vector; or, in symbols, 0 + a = a. If then we agree to abridge generally (comp. 4) the expression 0+ a to + a, and if a still denote a vector, then + a, and + (+ a), &c., are other symbols for the same vector; and we have, as in algebra, the identities,

− (− a) = + a, + (− a) = − (+ a) = − a, (+ a) + (− a) = 0, &c.

SECTION 3.-On Sums of three or more Vectors. 8. The sum of three given vectors, a, ß, y, is next defined to be that fourth vector,

d=y+ (B+a), or briefly, d=y+B+a,

which is obtained by adding the third to the sum of the first and second; and in like manner the sum of any number of vectors is formed by adding the last to the sum of all that

precede it: also, for any four vectors, a, ß, y, d, the sum 8+(y+B+ a) is denoted simply by 8+y+B+a, without parentheses, and so on for any number of summands.

9. The sum of any number of successive vectors, AB, BC, CD, is thus the line AD, which is

drawn from the origin a of the first,


Fig. 9.

to the term D of the last; and because, when there are three such vectors, we can draw (as in Fig. 9) the two diagonals AC, BD of the (plane or gauche) quadrilateral ABCD, and may then at pleasure regard AD, either as the sum of ab, bd, or as the sum of AC, CD, we are allowed to establish the following general formula of association, for the case of any three summand lines, a, ẞ, y:

(y + B) + a = y + (ẞ + a) = y + ẞ +a;

by combining which with the formula of commutation (Art. 7), namely, with the equation,

a+ B = B+ a,

which had been previously established for the case of any two such summands, it is easy to conclude that the Addition of Vectors is always both an Associative and a Commutative Operation. In other words, the sum of any number of given vectors has a value which is independent of their order, and of the mode of grouping them; so that if the lengths and directions of the summands be preserved, the length and direction of the sum will also remain unchanged: except that this last direction may be regarded as indeterminate, when the length of the sumline happens to vanish, as in the case which we are about to consider.

10. When any n summand-lines, AB, BC, CA, or AB, BC, CD, DA, &c., arranged in any one order, are the n successive sides of a triangle ABC, or of A

a quadrilateral ABCD, or of any other


Fig. 10.


closed polygon, their sum is a null line, aa; and conversely,




when the sum of any given system of n vectors is thus equal to zero, they may be made (in any order, by transports without rotation) the n successive sides of a closed polygon (plane or gauche). Hence, if there be given any such polygon (P), suppose a pentagon ABCDE, it is possible to construct another closed polygon (P'), such as A'B'C'D'E', with an arbitrary initial point a', but with the same number of sides, A'B',.. E'A', which new sides shall be equal (as vectors) to the old sides AB,.. EA, taken in any arbitrary order. For example, if we draw four successive vectors, as follows,

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and then complete the new pentagon by drawing the line E'A', tha this closing side of the second figure (P') will be equal to the remaining side DE of the first figure (P).

11. Since a closed figure ABC.. is still a closed one, when all its points are projected on any assumed plane, by any system of parallel ordinates (although the

area of the projected figure A'B'C'...
may happen to vanish), it follows that
if the sum of any number of given
a, 3,


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zero, and if we project them all on any one plane by parallel lines drawn from their extre

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Fig. 11.

mities, the sum of the projected vec-
tors a', B', y,.. will likewise be null; A
so that these latter vectors, like the
former, can be so placed as to become the successive sides of a
closed polygon, even if they be not already such. (In Fig. 11,
A"B"C" is considered as such a polygon, namely, as a triangle
with evanescent area; and we have the equation,

as well as

A"B" + B"C" + C"A"= 0,

A'B' + B'C' + C'A' = 0, and AB + BC + CA = 0.)

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