Question

If $a, b, c$ are the direction ratios of a line and $l, m, n$ are the direction cosines of the line, then

• A. $I=\pm a\left(a^2+b^2+c^2\right), m=\pm b\left(a^2+b^2+c^2\right)$, $n=\pm c\left(a^2+b^2+c^2\right)$
• B. $l=\pm \frac{a}{a^2+b^2+c^2}, m=\pm \frac{b}{a^2+b^2+c^2}, n=\pm \frac{c}{a^2+b^2+c^2}$
• C. $ l=\pm \frac{\sqrt{a}}{\sqrt{a^2+b^2+c^2}}, m=\pm \frac{\sqrt{b}}{\sqrt{a^2}+b^2+c^2}, n=\pm \frac{\sqrt{c}}{\sqrt{a^2+b^2+c^2}} $
• D. $ I=\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}}$

Answer 1

Answer: (D)

Let $a, b$ and $c$ be direction ratios of a line and $I, m$ and $n$ be the direction cosines (DC's) of the line.
Then, $\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=k$ (say), $k$ being a constant.
Therefore, $I=a k, m=b k, n=c k$...(i)
But $\quad l^2+m^2+n^2=1$
Therefore, $k^2\left(a^2+b^2+c^2\right)=1$
or $k=\pm \frac{1}{\sqrt{a^2+b^2+c^2}}$
Hence, from Eq. (i), the DC's of the line are
$ I=\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, depending on the desired sign of $k$, either a positive or a negative sign is to be taken for $l, m$ and $n$.