(1) Direction cosines : If \[\alpha ,\,\,\beta ,\,\,\gamma \] be the angles which a given directed line makes with the positive direction of the \[x,\,\,y,\,\,z\] co-ordinate axes respectively, then \[\cos \alpha ,\,\cos \beta ,\,\cos \gamma \] are called the direction cosines of the given line and are generally denoted by \[l,\,m,\,n\] respectively.
Thus, \[l=\cos \alpha ,\,\,m=\cos \beta \] and \[n=\cos \gamma ,\,\,{{l}^{2}}+{{m}^{2}}+{{n}^{2}}=1\].
By definition, it follows that the direction cosine of the axis of \[x\] are respectively \[\cos {{0}^{o}},\,\,\cos {{90}^{o}},\,\,\cos {{90}^{o}}\] i.e., \[(1,\,\,0,\,\,0)\]. Similarly direction cosines of the axes of \[y\] and \[z\] are respectively \[(0,\,\,1,\,\,0)\] and \[(0,\,\,0,\,\,1)\].
(2) Direction ratios: If \[a,b,c\] are three numbers proportional to direction cosines \[l,\,\,m,\,\,n\] of a line, then \[a,\,\,b,\,\,\,c\] are called its direction ratios. They are also called direction numbers or direction components.
Hence by definition,
\[l=\pm \frac{a}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}\],\[m=\pm \frac{b}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}\],\[n=\pm \frac{c}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}\]
where
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