Page:Elementary Text-book of Physics (Anthony, 1897).djvu/26

displacements are combined as if they were successive displacements. Representing them both by straight lines, of which the length measures the amount of the displacement, and the direction the direction of the displacement (Fig. 5), we apply the initial point of the second of these lines to the final point of the first and join the initial point of the first to the final point of the second. The line thus drawn is the resultant of the simultaneous displacements. The two displacements of which the resultant is thus obtained are called the components.

13. Vector Addition and Subtraction.— Any concept which is completely described when its magnitude and direction are given is called a vector. The sum of two vectors is the vector equivalent to them both. It is obtained by the rule just given for the composition of two displacements, or by the following equivalent rule: Draw from any point the two straight lines which represent the vectors, and upon them construct a parallelogram; the diagonal of this parallelogram, drawn from the point of origin, is the resultant vector or the vector sum. Thus $OC$ (Fig. 5) is the resultant of $OA$ and $OB$. This construction is called the parallelogram construction or the parallelogram law. If more than two vectors are to be added, the resultant of two of them may be added to the third, the resultant thus obtained to the fourth, and so on until all the vectors have been combined. This addition is more easily made by drawing the vectors in succession, so that they form the sides of a polygon (Fig. 6), the initial point of each vector coinciding with the final point of the one preceding it. In general this polygon is not closed, and the line required to close it, drawn from the initial point of the first vector to the final point of the last, is the sum of the vectors. This construction is called the polygon construction or the polygon law.

The difference of two vectors is the vector which added to one of the two will give the other. It is obtained by drawing from a given point the lines representing the vectors, and drawing a line