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.