Question

If $\left(\frac{1}{2},\frac{1}{3},n\right)$ are the direction cosines of a line, then the value of $n$ is

• A. $\frac{\sqrt{23}}{6}$
• B. $\frac{23}{6}$
• C. $\frac{2}{3}$
• D. $\frac{3}{2}$

Answer 1

The correct option is A

$\frac{\sqrt{23}}{6}$

Explanation for correct option:

Given, Direction cosine of a line $\left(\frac{1}{2},\frac{1}{3},n\right)$.

We know that relation between direction cosine is $\sqrt{{\mathbf{l}}^{\mathbf{2}}\mathbf{+}{\mathbf{m}}^{\mathbf{2}}\mathbf{+}{\mathbf{n}}^{\mathbf{2}}}\mathbf{=}\mathbf{1}$

$\Rightarrow$$\sqrt{{\left(\frac{1}{2}\right)}^2+{\left(\frac{1}{3}\right)}^2+n^2}=1$

Squaring both sides.

$\Rightarrow$$\frac{1}{4}+\frac{1}{9}+n^2=1$

$\Rightarrow$$n^2=1-\frac{13}{36}$

$\Rightarrow$$n^2=\frac{23}{36}$

$\Rightarrow$$n=\frac{\sqrt{23}}{6}$

Hence, correct answer is option $\left(A\right)$