Question

If a line has direction ratios 2, -1, -2, then what are its direction cosines?

(a) $\pm \dfrac{2}{3},\mp \dfrac{1}{3},\mp \dfrac{2}{3}$
(b) $\mp \dfrac{1}{3},\pm \dfrac{2}{3},\mp \dfrac{1}{3}$
(c) $\mp \dfrac{2}{3},\mp \dfrac{2}{3},\mp \dfrac{2}{3}$
(d) $\pm \dfrac{1}{3},\pm \dfrac{1}{3},\pm \dfrac{1}{3}$

Answer 1

Firstly we should know that direction ratios of any particular line are the numbers which are proportional to direction cosines. Generally, direction cosines are represented by l, m and n respectively made with X – axis, Y – axis and Z – axis respectively whereas a, b, c are their respective direction ratios. We know that the direction cosines in terms of direction ratios are represented as
$l=\dfrac{a}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}},\text{ }m=\dfrac{b}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}},\text{ }n=\dfrac{c}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}$
Hence, we will have to replace the values of direction ratios a, b and c in the given formula with the data given in the question and solve them to get the required direction cosines.

Complete step by step answer:
Here, the values of the direction ratios are given as $a=2,\text{ b = }-1\text{, c =}-2$ according to our given question. Now, we will replace the value of a, b and c in the above formula with 2, -1 and -2 respectively and find the required values of direction cosines.
Replacing the value of a with 2, b with -1 and c with -2 in first formula to get the value of l, we get,
$\Rightarrow l=\dfrac{2}{\sqrt{{{2}^{2}}+{{\left( -1 \right)}^{2}}+{{\left( -2 \right)}^{2}}}}\text{ }$
$\Rightarrow l=\dfrac{2}{\sqrt{4+1+4}}\text{ }$
$\Rightarrow l=\dfrac{2}{\sqrt{9}}$
$\Rightarrow l=\dfrac{2}{\pm 3}$
$\therefore \text{l=}\pm \dfrac{2}{3}\text{ }$
Hence, when $a=2,\text{ b = }-1\text{, c =}-2$, the value of l is $\pm \dfrac{2}{3}.$
Similarly, substituting the value of a with 2, b with -1 and c with -2 in second formula to get the value of m, we get,
$\Rightarrow m=\dfrac{\left( -1 \right)}{\sqrt{{{2}^{2}}+{{\left( -1 \right)}^{2}}+{{\left( -2 \right)}^{2}}}}\text{ }$
$\Rightarrow m=\dfrac{\left( -1 \right)}{\sqrt{4+1+4}}\text{ }$
$\Rightarrow m=\dfrac{\left( -1 \right)}{\sqrt{9}}$
$\Rightarrow m=\dfrac{\left( -1 \right)}{\pm 3}$
$\therefore m=\mp \dfrac{1}{3}\text{ }$
Hence, when $a=2,\text{ b = }-1\text{, c =}-2$, the value of m is $\mp \dfrac{1}{3}\text{. }$

Again, substituting the value of a with 2, b with -1 and c with -2 in third formula to get the value of n, we get,
$\Rightarrow n=\dfrac{\left( -2 \right)}{\sqrt{{{2}^{2}}+{{\left( -1 \right)}^{2}}+{{\left( -2 \right)}^{2}}}}\text{ }$
$\Rightarrow n=\dfrac{\left( -2 \right)}{\sqrt{4+1+4}}\text{ }$
$\Rightarrow n=\dfrac{\left( -2 \right)}{\sqrt{9}}$
$\Rightarrow n=\dfrac{\left( -2 \right)}{\pm 3}$
$\therefore n=\mp \dfrac{2}{3}\text{ }$
Hence, when $a=2,\text{ b = }-1\text{, c =}-2$, the value of n is $\mp \dfrac{2}{3}.$

So, the correct answer is “Option A”.

Note: Students usually make the mistake of finding only the value of first direction cosine, l and then choosing whichever option has that value without considering the values of other direction cosines, m and n. Therefore to avoid making such errors, firstly students must find the values of all direction cosines, l, m and n and then choose the correct option according to the calculated answer among all other options.