The direction cosines of a line equally inclines to three mutually perpendicular lines having D.C.'s
as `(l_(1),m_(1),n_(1)),(l_(2),m_(2),n_(2)),(l_(3),m_(3),n_(3))` are

Options

A. `l_(1)+l_(2)+l_(3),m_(1)+m_(2)+m_(3),n_(1)+n_(2)+n_(3)`
B. `(l_(1)+l_(2)+l_(3))/sqrt3,(m_(1)+m_(2)+m_(3))/sqrt3,(n_(1)+n_(2)+n_(3))/sqrt3`
C. `(l_(1)+l_(2)+l_(3))/3,(m_(1)+m_(2)+m_(3))/3,(n_(1)+n_(2)+n_(3))/3`
D. none of these

Answer

To find the direction cosines of a line that equally inclines to three mutually perpendicular lines, we can follow these steps:

### Step 1: Understand the Direction Cosines
The direction cosines of a line are represented as \( (l, m, n) \). For three mutually perpendicular lines with direction cosines \( (l_1, m_1, n_1) \), \( (l_2, m_2, n_2) \), and \( (l_3, m_3, n_3) \), we know that the following conditions hold:
1. \( l_1^2 + m_1^2 + n_1^2 = 1 \)
2. \( l_2^2 + m_2^2 + n_2^2 = 1 \)
3. \( l_3^2 + m_3^2 + n_3^2 = 1 \)
4. The dot products of the direction cosines of any two lines are zero:
- \( l_1l_2 + m_1m_2 + n_1n_2 = 0 \)
- \( l_2l_3 + m_2m_3 + n_2n_3 = 0 \)
- \( l_1l_3 + m_1m_3 + n_1n_3 = 0 \)

### Step 2: Set Up the Equations
Since the line is equally inclined to the three mutually perpendicular lines, we can express the conditions mathematically. The line's direction cosines \( (l, m, n) \) must satisfy:
- \( l = m = n \)

### Step 3: Use the Condition of Equally Inclined
Since the line is equally inclined to the three lines, we can use the fact that the direction cosines can be expressed as:
\[ l^2 + m^2 + n^2 = 1 \]
Substituting \( l = m = n \) gives:
\[ 3l^2 = 1 \implies l^2 = \frac{1}{3} \implies l = \pm \frac{1}{\sqrt{3}} \]
Thus, we have:
\[ l = m = n = \frac{1}{\sqrt{3}} \text{ or } -\frac{1}{\sqrt{3}} \]

### Step 4: Conclusion
The direction cosines of the line that equally inclines to the three mutually perpendicular lines are:
\[ \left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \text{ or } \left( -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} \right) \]