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Addition and Subtraction of Vectors, or Geometric Addition and Subtraction.
1. A Vector is the representative of transference through a given distance in a given direction.
Thus, if A, B are any two points, vector AB implies a translation from A to B.
A vector may be represented geometrically by a right line, whose length denotes the distance over which transference takes place, and whose direction denotes the direction of the transference. In thus designating a vector, the direction is indicated by the order of the letters.
Thus, AB (Fig. 1) denotes transference from A to B, and BA from в to A.
Retaining the algebraic signification of the signs and, if AB denotes motion from A to B, then -AB will denote motion from B to A, and
AB= -BA, -AB BA
Hence, the effect of a minus sign before a vector is to reverse its direction.
The conception of a vector, therefore, implies that of its two elements, distance and direct on; it was first defined as a directed right line. It is now applied more generally to all quantities. determined by magnitude and direction. Thus, force, the path
of a moving body, velocity, an electric current, etc., are vector quantities.
Analytically, vectors are represented by the letters of the Greek alphabet, a, ẞ, y, etc.
2. It follows, from the definition of a vector, that all lines which are equal and parallel may be represented by the same vector symbol with like or unlike signs. If equal and drawn in the same direction, they will have the same sign. Hence an equality between two vectors implies equality in distance with the same direction.
Thus, if AB (Fig. 2), CD, BE, EF
and IG are equal and drawn in the same direction, they may be represented by the same vector symbol, and
3. It follows also from the definition of a vector that, if vectors are not parallel, they cannot be represented by the same vector symbol.
Thus, if the point A (Fig. 3) move over the right line AB, from A to B, and then over the right line BC, from в to c, and
AB = a, BC must be denoted by some other symbol, as ß.
The result of these two successive translations of the point a is the same as that of the single and direct translation AC=7, from a to c; in either case A is found at the extremity of the diagonal of the parallelogram of which AB and BC are the sides. This combination of successive translations is called addition, and is written in the ordinary way,
This expression would be absurd if the symbols denoted magnitudes only. It means that transference from A to B, followed
by transference from в to C, is equivalent to transference from does not therefore denote a numerical ad
A to c. The sign
dition, or the sign an equality between magnitudes. It is, however, called an equation, and read, as usual, “a plus ẞ is equal to y." This kind of addition is called geometric addition.
4. If the point a (Fig. 3), instead of moving over the sides AB, BC of the parallelogram ABCD, had moved in succession over the other two sides, AD and DC, the result would still have been the same as that of the single translation over the diagonal Ac. But since AB and BC are equal in length to DC and AD respectively, and are drawn in the same direction, we have (Art. 2)
AB DC and BC= AD,
and if the first two translations are represented by AB and BC, the second two may be represented by BC and ab, or
Hence the operation of vector addition is commutative, or the sum of any number of given vectors is independent of their order.
5. If the point A (Fig. 4) move in succession over the three edges AB, BC, CG of a parallelopiped, we have
and the operation of vector addition is associative, or the sum of any number of given vectors is independent of the mode of grouping them.
a term may be transposed from one member to another in a vector equation by changing its sign.
Also, in every triangle, any side may be considered as the sum or difference of the other two, depending upon their directions as vectors. Thus (Fig. 3) @
2 y― B=a,
It is to be observed that no one direction is assumed as positive, as in Cartesian Geometry. The only assumption is that opposite directions shall have opposite signs. The results must, of course, be interpreted in accordance with the primitive assumptions. Thus, had we assumed BA=a (Fig. 3), y and ß being as before, then
B-a = 7,
7. If two vectors having the same direction be added together, the sum will be a vector in the same direction. If the vectors be also equal in length, the length of the vector sum will be the sum of their lengths. If n vectors, of equal length and drawn in the same direction, be added together, the sum will be the product of one of these vectors by n, or a vector having the same direction and whose length is n times the common length. If then (Fig. 2)
AF XAB = XCD = xα,
where A, B and F are in the same straight line, CD AB, and x is a positive whole number, a expresses the ratio of the lengths of AF and a. From the case in which x is an integer we pass, by the usual reasoning, to that in which it is fractional or incommensurable. Vectors, then, in the same direction, have the same ratio as the corresponding lengths.
If AB a be assumed as the unit vector, then
in which m is a positive numerical quantity and is called the Tensor. It is the ratio of the length of the vector ma to that of the unit vector a, or the numerical factor by which the unit vector is multiplied to produce the given vector.
Any vector, as ẞ, may be written in general notation
In this notation, Tẞ (read "tensor of ẞ") is the numerical factor which stretches the unit vector so that it shall have the proper length; hence its name, tensor. It is, strictly speaking, an abstract number without sign, but, to distinguish between it and the negative of algebra, it may be said to be always posi- for tive. Uß (read "versor of ẞ") is the unit vector having the direction of ß; the reason for the name versor will appear later.
T and U are also general symbols of operation. Written before an expression, they denote the operations of taking the tensor and versor, respectively. Thus, if the length of ẞ is n times that of the unit vector,
where T denotes the operation of taking the stretching factor, i.e. the tensor. While
U(B) = Uß
indicates the operation of taking the unit vector, that is, of reducing a vector ẞ to its unit of length without changing its direction.
8. If BC (Fig. 5) be any vector, and BA = YBC, then
-BA AB= - YBC;
and, in general, if BA and BC be
any two real vectors, parallel and
of unequal length, we may always conceive of a coefficient y which shall satisfy the equation