Page:Elementary Text-book of Physics (Anthony, 1897).djvu/28

Fig. 7The composition of vectors is often conveniently effected by resolving them in this way along the three coordinate axes; their components along each of these axes may then be added algebraically, and the vector obtained by combining the three sums is the required resultant vector. Thus if the vectors $R_{1},R_{2}...R_{n}$ be given, making angles with the $x,y,z$-axes of which the cosines are $\lambda _{1},\lambda _{2}...\lambda _{n},\mu _{1},\mu _{2}...\mu _{n},\nu _{1},\nu _{2}...\nu _{n},$ respectively, the sums of the components of these vectors along the axes are

$X=R_{1}\lambda _{1}+R_{2}\lambda _{2}+...+R_{n}\lambda _{n};$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$ $(1)$
$Y=R_{1}\mu _{1}+R_{2}\mu _{2}+...+R_{n}\mu _{n};$
$Z=R_{1}\nu _{1}+R_{2}\nu _{2}+...+R_{n}\nu _{n}.$

The resultant vector is

$R={\sqrt {X^{2}+Y^{2}+Z^{2}}},$

and its direction cosines are

${\frac {X}{R}},\,{\frac {Y}{R}},\,{\frac {Z}{R}},$

respectively.

When only two vectors are given, they may be resolved along two axes in the plane of the vectors. In this case, if the angles made by the vectors $R_{1},R_{2}$ with the x-axis be $\phi ,\vartheta$, respectively, (Fig. 7,) the component sums are

$X=R_{1}\cos {\phi }+R_{2}\cos {\vartheta },$	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$ $(2)$
$X=R_{1}\sin {\phi }+R_{2}\sin {\vartheta },$

The resultant vector is $R={\sqrt {X^{2}+Y^{2}}}$, and the angle $\psi$ which it makes with the x-axis is given by $\cos {\psi }={\frac {X}{R}}$ or $\tan {\psi }={\frac {Y}{X}}\cdot$

15. Description of Motion.— If we observe a system of points in motion, we perceive not only the displacements of the points, but also that these displacements are in some way connected with the time required for their accomplishment. If we know the law of this connection, we may describe the motion at any desired instant, by the aid of certain derived concepts, which are now to be studied.

If a variable quantity be a function of the time, it is usual in