# Current Affairs JEE Main & Advanced

## Direction Cosines and Direction Ratios

Category : JEE Main & Advanced

(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., $l=\frac{{{x}_{2}}-{{x}_{1}}}{PQ},\,m=\frac{{{y}_{2}}-{{y}_{1}}}{PQ},\,n=\frac{{{z}_{2}}-{{z}_{1}}}{PQ}$.

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