Question

If the length of a vector is $21$ and direction ratios be $2,-3,6$ then its direction cosines are.

• A. $\frac{2}{21},-\frac{1}{7},\frac{2}{7}$
• B. $\frac{2}{7},-\frac{3}{7},\frac{6}{7}$
• C. $\frac{2}{7},\frac{3}{7},\frac{6}{7}$
• D. None of these.

Answer 1

The correct option is B

$\frac{2}{7},-\frac{3}{7},\frac{6}{7}$

Explanation for the correct option:

Find direction cosine:

The given direction ratios are $2,-3,6$.

Let us assume that, $a=2,b=-3,c=6$.

Thus the direction cosines of the vector can be given by, $l=\frac{a}{\sqrt{a^2+b^2+c^2}}$, $m=\frac{b}{\sqrt{a^2+b^2+c^2}}$ and $n=\frac{c}{\sqrt{a^2+b^2+c^2}}$.

Firstly, $\sqrt{a^2+b^2+c^2}=\sqrt{2^2+{\left(-3\right)}^2+6^2}$

$$\begin{gathered} =\sqrt{4+9+36} \\ =\sqrt{49} \\ =7 \end{gathered}$$

So, $l=\frac{a}{\sqrt{a^2+b^2+c^2}}$$\Rightarrow l=\frac{2}{7}$

$m=\frac{b}{\sqrt{a^2+b^2+c^2}}$$\Rightarrow m=-\frac{3}{7}$

$n=\frac{c}{\sqrt{a^2+b^2+c^2}}$$\Rightarrow n=\frac{6}{7}$

Therefore, the direction cosines are $\frac{2}{7},-\frac{3}{7},\frac{6}{7}$.

Hence, option(B) is the correct option.