JEE Main & Advanced Mathematics Three Dimensional Geometry Direction Cosines and Direction Ratios

(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 the sign should be taken all positive or all negative.

Direction ratios are not unique, whereas d.c.’s are unique.     i.e., \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\ne 1\].

(3) D.c.’s and d.r.’s of a line joining two points : The direction ratios of line PQ joining \[P({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})\] and \[Q({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})\] are \[{{x}_{2}}-{{x}_{1}}=a\], \[{{y}_{2}}-{{y}_{1}}=b\] and \[{{z}_{2}}-{{z}_{1}}=c\], (say).

Then direction cosines are, 

\[l=\frac{({{x}_{2}}-{{x}_{1}})}{\sqrt{\sum {{({{x}_{2}}-{{x}_{1}})}^{2}}}},\,\text{ }m=\frac{({{y}_{2}}-{{y}_{1}})}{\sqrt{\sum {{({{x}_{2}}-{{x}_{1}})}^{2}}}},\,\text{ }n=\frac{({{z}_{2}}-{{z}_{1}})}{\sqrt{\sum {{({{x}_{2}}-{{x}_{1}})}^{2}}}}\]

i.e., \[l=\frac{{{x}_{2}}-{{x}_{1}}}{PQ},\,m=\frac{{{y}_{2}}-{{y}_{1}}}{PQ},\,n=\frac{{{z}_{2}}-{{z}_{1}}}{PQ}\].