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an 
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having always to be satisfied. 
Two vectors are said to be perpendicular to each other, if (GS, ©) = 90°. 
If the angle is 180°, the vectors have opposite directions. This 
will sometimes be expressed by saying that the two have the same 
direction-constants, but that one value is positive and the other negative. 
§ 5. Multiplying a vector © by a positive or negative number & 
means, that each element is multiplied by &, and that the products 
are taken as the elements of a new vector, which we shall indicate 
by £6. 
Two vectors ©, and G, are said to be compounded with each 
other, if any two corresponding elements are added algebraically, 
and the sums thus obtained are taken as the elements of a new 
vector. This is called the resultant or the sum of the two vectors, 
and represented by S + Ss; it may again be decomposed into the 
components S, and Gy. 
There are a number of theorems, closely corresponding to those 
in the theory of ordinary vectors. We need only mention some of them. 
If Cin lor Se pina Riles It Se A 
and if A be an arbitrarily chosen direction in the system, i. e. the 
direction of some vector in the system, we shall have 
Sj cos (Sj ,h) + 93 cos (Gg ‚h) = Sz cos (Ss , h). 
From this it appears that, as soon as two of the vectors ©), Gg, Ss; 
are perpendicular to the direction kh, the third will likewise be so. 
It may further be shown that a given vector © may always be 
decomposed into one component having a given direction 4 (or preci- 
sely the opposite direction) and a second component, perpendicular 
to h. This decomposition can be effected in only one way, the value 
of the first component being Scos(S,h). This may be positive or 
negative; in one case the component has the direction A, in the other 
it has the opposite direction. 
The value of the component along 4 is also called the projection 
of © on the direction A. 
By the scalar product of the vectors ©) and ©, we understand 
the expression 
S, Sg cos (Sj, Sa) » 
for which the sign (€,.©,) will be used. 
It is also to be remarked that, in the case of (7), 
SP EPS IDO EN Fi eee een 
and that we may regard the formula 
CT 
