Page:Elementary Text-book of Physics (Anthony, 1897).djvu/27

§ 14] from the final point of the subtrahend to the final point of the minuend. This line represents the vector difference of the two vectors. Thus $$AC$$ (Fig. 5) is the difference between $$OC$$ and $$OA$$. The same difference may be obtained by the following method: If two lines, equal in length, be drawn in opposite directions, they represent two vector quantities which have the same magnitude but are affected with opposite signs. If, therefore, a vector be given which is to be subtracted from another, it may be replaced by a vector of the same magnitude having the opposite direction, and the resultant obtained by adding this vector to the one which serves as the minuend is the difference of the two given vectors.

14. Resolution and Composition of Vectors.— It is in many cases convenient to obtain component vectors which are equivalent to a given vector. If one component be completely given, the other is obtained by vector subtraction. If two components be desired, and their directions be given so that they and the original vector are in the same plane, their magnitudes may be determined by drawing from a common origin lines of indefinite length in the given directions, drawing from the same origin the line representing the given vector, and drawing from its final point lines parallel to the given directions. The sides of the parallelogram thus constructed represent the component vectors in these given directions.

If three components be desired in three given directions not in the same plane, and so placed that the given vector does not lie in a plane containing any two of these directions, they may be found by constructing upon lines drawn in these directions a parallelepiped of which the diagonal is the given vector. This construction is most frequently used when the three directions are at right angles to one another. Representing the angles between them and the direction of the given vector by $$\alpha, \beta, \gamma$$ the component vectors are proportional to $$\cos{\alpha}, \cos{\beta}, \cos{\gamma}$$. If these three directions be the directions of the axes of a system of rectangular coordinates, these cosines are called the direction cosines of the vector.