Question

If the direction ratios of a line are $ 1, -3, 2, $ then the direction cosines of the line are :

• A. $ \frac{1}{\sqrt{14}},\,\,\frac{-3}{\sqrt{14}},\,\,\frac{2}{\sqrt{14}} $
• B. $ \frac{1}{\sqrt{14}},\,\,\frac{2}{\sqrt{14}},\,\,\frac{3}{\sqrt{14}} $
• C. $ \frac{-1}{\sqrt{14}},\,\,\frac{3}{\sqrt{14}},\,\frac{-2}{\sqrt{14}} $
• D. $ \frac{-1}{\sqrt{14}},\,\,\frac{-2}{\sqrt{14}},\,\,\frac{-3}{\sqrt{14}} $

Answer 1

Answer: (A)

If $a, b$ and $c$ are the direction ratios of a line, then the direction cosines of a line are
$l=\frac{a}{\sqrt{a^{2}+b^{2}+c^{2}}}$,
$m=\frac{b}{\sqrt{a^{2}+b^{2}+c^{2}}} $
$n=\frac{c}{\sqrt{a^{2}+b^{2}+c^{2}}}$
Given direction ratios are $1,-3$ and $2 $.
$\therefore $ Direction cosines are
$l=\frac{1}{\sqrt{1^{2}+(-3)^{2}+2^{2}}}$,
$ m=-\frac{3}{\sqrt{1^{2}+(-3)^{2}+2^{2}}}$
$n=\frac{2}{\sqrt{1^{2}+(-3)^{2}+(2)^{2}}}$
i.e ., $l=\frac{1}{\sqrt{14}}, m=-\frac{3}{\sqrt{14}}, n=\frac{2}{\sqrt{14}}$
In any line has only one direction cosine but the direction ratios may be more than one.