What are the direction cosenes of a line which is equally inclined to the positive directions of the axes

Options

A. `ltlt1/sqrt3,1/sqrt3,1/sqrt3gtgt`
B. `ltlarr1/sqrt3,1/sqrt3,1/sqrt3gtgt`
C. `ltlarr1/sqrt3,-1/sqrt3,1/sqrt3gtgt`
D. `ltlt1/3,1/3,1/3gtgt`

Answer

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

### Step 1: Understand Direction Cosines
Direction cosines are the cosines of the angles that a line makes with the coordinate axes. If a line makes angles \( \alpha, \beta, \) and \( \gamma \) with the x, y, and z axes respectively, then the direction cosines \( l, m, n \) are given by:
- \( l = \cos(\alpha) \)
- \( m = \cos(\beta) \)
- \( n = \cos(\gamma) \)

### Step 2: Equally Inclined Condition
Since the line is equally inclined to the positive directions of the axes, we have:
- \( \alpha = \beta = \gamma \)

This means that the direction cosines are equal:
- \( l = m = n \)

### Step 3: Use the Property of Direction Cosines
We know from the property of direction cosines that:
\[ l^2 + m^2 + n^2 = 1 \]
Substituting \( l = m = n \) into this equation gives:
\[ l^2 + l^2 + l^2 = 1 \]
or
\[ 3l^2 = 1 \]

### Step 4: Solve for \( l \)
Now, we can solve for \( l \):
\[ l^2 = \frac{1}{3} \]
Taking the square root gives:
\[ l = \frac{1}{\sqrt{3}} \]

### Step 5: Find Direction Cosines
Since \( l = m = n \), we have:
- \( l = m = n = \frac{1}{\sqrt{3}} \)

### Conclusion
Thus, the direction cosines of the line which is equally inclined to the positive directions of the axes are:
\[ \left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \]