If a line is equally inclined with the coordinate axes, then its direction cosines are

Options

A. `pm lt 1,1,1, gt`
B. `pm lt (1)/(3) , (1)/(3) , (1)/(3) gt`
C. `pm lt (1)/( sqrt3) , (1)/( sqrt3) , (1)/( sqrt3) gt`
D. `pm lt (1)/( sqrt3) , (-1)/( sqrt3) , (-1)/( sqrt3) gt`

Answer

To find the direction cosines of a line that is equally inclined with the coordinate axes, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Direction Cosines**:
- The direction cosines of a line are the cosines of the angles that the line makes with the coordinate axes (x, y, z). Let these angles be denoted as α, β, and γ respectively.

2. **Condition for Equally Inclined Line**:
- Since the line is equally inclined to the coordinate axes, we have:
\[ \cos \alpha = \cos \beta = \cos \gamma \]
- Let's denote this common value as \( k \). Thus, we can write:
\[ \cos \alpha = \cos \beta = \cos \gamma = k \]

3. **Using the Direction Cosine Condition**:
- The sum of the squares of the direction cosines must equal 1:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]
- Substituting \( k \) into this equation gives:
\[ k^2 + k^2 + k^2 = 1 \]
- This simplifies to:
\[ 3k^2 = 1 \]

4. **Solving for k**:
- Dividing both sides by 3, we find:
\[ k^2 = \frac{1}{3} \]
- Taking the square root of both sides, we get:
\[ k = \pm \frac{1}{\sqrt{3}} \]

5. **Direction Cosines**:
- Since \( k = \cos \alpha = \cos \beta = \cos \gamma \), the direction cosines of the line are:
\[ \left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \text{ or } \left( -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} \right) \]

### Final Answer:
The direction cosines of the line that is equally inclined with the coordinate axes are:
\[ \left( \pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}} \right) \]