If the direction cosines of two lines are `(l_(1), m_(1), n_(1))` and
`(l_(2), m_(2), n_(2))` and the angle between them is `theta` then
`l_(1)^(2)+m_(1)^(2)+n_(1)^(2)=1=l_(2)^(2)+m_(2)^(2)+n_(2)^(2)`
and costheta` = l_(1)l_(2)+m_(1)m_(2)+n_(1)n_(2)`
If `l_(1)=1/sqrt(3), m_(1)=1/sqrt(3)` then the value of `n_(1)` is equal to

Options

A. `pm 1/sqrt(3)`
B. `+ 1/sqrt(3)`
C. ` -1/sqrt(3)`
D. 0

Answer

To find the value of \( n_1 \) given the direction cosines \( l_1 \) and \( m_1 \), we can follow these steps:

### Step 1: Use the equation for direction cosines
The equation for direction cosines states that:
\[ l_1^2 + m_1^2 + n_1^2 = 1 \]
We know that \( l_1 = \frac{1}{\sqrt{3}} \) and \( m_1 = \frac{1}{\sqrt{3}} \).

### Step 2: Substitute the known values
Substituting the values of \( l_1 \) and \( m_1 \) into the equation:
\[ \left(\frac{1}{\sqrt{3}}\right)^2 + \left(\frac{1}{\sqrt{3}}\right)^2 + n_1^2 = 1 \]

### Step 3: Calculate \( l_1^2 \) and \( m_1^2 \)
Calculating \( l_1^2 \) and \( m_1^2 \):
\[ \frac{1}{3} + \frac{1}{3} + n_1^2 = 1 \]

### Step 4: Simplify the equation
Combine the fractions:
\[ \frac{2}{3} + n_1^2 = 1 \]

### Step 5: Isolate \( n_1^2 \)
Now, isolate \( n_1^2 \):
\[ n_1^2 = 1 - \frac{2}{3} \]
\[ n_1^2 = \frac{1}{3} \]

### Step 6: Solve for \( n_1 \)
Taking the square root of both sides gives:
\[ n_1 = \pm \frac{1}{\sqrt{3}} \]

### Final Answer
Thus, the value of \( n_1 \) is:
\[ n_1 = \frac{1}{\sqrt{3}} \quad \text{or} \quad n_1 = -\frac{1}{\sqrt{3}} \]

---