Question

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

• A. $\dfrac{\sqrt{23}}{6}$
• B. $\dfrac{23}{36}$
• C. $\dfrac{2}{3}$
• D. $\dfrac{2}{3}$

Hint:

Direction cosines $(l, m, n)$ always satisfy $l^2 + m^2 + n^2 = 1$. Use this to find any unknown direction cosine.

Answer 1

The Correct Option is A

Step 1: Conceptual Understanding:
The sum of squares of direction cosines of any line equals 1: $l^2 + m^2 + n^2 = 1$.
Step 2: Explanation in Detail:
\[\frac{1}{4} + \frac{1}{9} + n^2 = 1 \Rightarrow n^2 = 1 - \frac{9+4}{36} = 1 - \frac{13}{36} = \frac{23}{36} \Rightarrow n = \frac{\sqrt{23}}{6}.\] Step 3: Therefore, Stating the Final Answer
$n = \dfrac{\sqrt{23}}{6}$.