If `1/2,1/3,n` are direction cosines of a line, then the value of `n` is

Options

A. `(sqrt(23))/6`
B. `23/6`
C. `2/3`
D. `1/6`

Explanation

To find the value of \( n \) when \( \frac{1}{2}, \frac{1}{3}, n \) are the direction cosines of a line, we can follow these steps:

### Step 1: Use the property of direction cosines
The sum of the squares of the direction cosines is equal to 1. Therefore, we can write the equation:
\[ l^2 + m^2 + n^2 = 1 \]
Substituting the given values:
\[ \left(\frac{1}{2}\right)^2 + \left(\frac{1}{3}\right)^2 + n^2 = 1 \]

### Step 2: Calculate the squares of the given direction cosines
Calculating the squares:
\[ \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
\[ \left(\frac{1}{3}\right)^2 = \frac{1}{9} \]
So, substituting these values into the equation gives:
\[ \frac{1}{4} + \frac{1}{9} + n^2 = 1 \]

### Step 3: Find a common denominator
The common denominator for \( \frac{1}{4} \) and \( \frac{1}{9} \) is 36. We can rewrite the fractions:
\[ \frac{1}{4} = \frac{9}{36}, \quad \frac{1}{9} = \frac{4}{36} \]
Thus, the equation becomes:
\[ \frac{9}{36} + \frac{4}{36} + n^2 = 1 \]

### Step 4: Combine the fractions
Combining the fractions gives:
\[ \frac{9 + 4}{36} + n^2 = 1 \]
This simplifies to:
\[ \frac{13}{36} + n^2 = 1 \]

### Step 5: Isolate \( n^2 \)
To isolate \( n^2 \), subtract \( \frac{13}{36} \) from both sides:
\[ n^2 = 1 - \frac{13}{36} \]
Converting 1 to a fraction with a denominator of 36:
\[ 1 = \frac{36}{36} \]
So, we have:
\[ n^2 = \frac{36}{36} - \frac{13}{36} = \frac{23}{36} \]

### Step 6: Solve for \( n \)
Taking the square root of both sides gives:
\[ n = \sqrt{\frac{23}{36}} = \frac{\sqrt{23}}{6} \]

Thus, the value of \( n \) is:
\[ \frac{\sqrt{23}}{6} \]

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