The direction cosines of a line equally inclined to three mutually perpendiclar lines having direction cosines as `l_(1),m_(1),n_(1),l_(2),m_(2),n_(2)` and `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))/sqrt(3), (m_(1)+m_(2)+m_(3))/sqrt(3), (n_(1)+n_(2)+n_(3))/sqrt(3)`
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

Explanation

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

### Step 1: Understand the Direction Cosines
The direction cosines of a line are defined as the cosines of the angles that the line makes with the coordinate axes. For three mutually perpendicular lines, we denote their direction cosines as:
- Line 1: \( (l_1, m_1, n_1) \)
- Line 2: \( (l_2, m_2, n_2) \)
- Line 3: \( (l_3, m_3, n_3) \)

### Step 2: Use the Property of Mutually Perpendicular Lines
Since the lines are mutually perpendicular, the dot products of their direction cosines are zero:
1. \( l_1 l_2 + m_1 m_2 + n_1 n_2 = 0 \)
2. \( l_2 l_3 + m_2 m_3 + n_2 n_3 = 0 \)
3. \( l_1 l_3 + m_1 m_3 + n_1 n_3 = 0 \)

### Step 3: Set Up the Equation for Equally Inclined Line
For a line that is equally inclined to these three lines, we need to find the direction cosines \( (l, m, n) \) such that:
\[ l^2 + m^2 + n^2 = 1 \]
This ensures that the direction cosines form a unit vector.

### Step 4: Express the Direction Cosines
Since the line is equally inclined to the three lines, we can express the direction cosines as:
\[ l = \frac{l_1 + l_2 + l_3}{\sqrt{3}}, \quad m = \frac{m_1 + m_2 + m_3}{\sqrt{3}}, \quad n = \frac{n_1 + n_2 + n_3}{\sqrt{3}} \]

### Step 5: Verify the Condition
We need to ensure that these values satisfy the condition \( l^2 + m^2 + n^2 = 1 \):
\[ \left( \frac{l_1 + l_2 + l_3}{\sqrt{3}} \right)^2 + \left( \frac{m_1 + m_2 + m_3}{\sqrt{3}} \right)^2 + \left( \frac{n_1 + n_2 + n_3}{\sqrt{3}} \right)^2 = 1 \]
This simplifies to:
\[ \frac{(l_1 + l_2 + l_3)^2 + (m_1 + m_2 + m_3)^2 + (n_1 + n_2 + n_3)^2}{3} = 1 \]
Thus, we have:
\[ (l_1 + l_2 + l_3)^2 + (m_1 + m_2 + m_3)^2 + (n_1 + n_2 + n_3)^2 = 3 \]

### Step 6: Final Direction Cosines
Thus, the direction cosines of the line that is equally inclined to the three mutually perpendicular lines are:
\[ \left( \frac{l_1 + l_2 + l_3}{\sqrt{3}}, \frac{m_1 + m_2 + m_3}{\sqrt{3}}, \frac{n_1 + n_2 + n_3}{\sqrt{3}} \right) \]