The sum of the direction cosines of a line which makes equal angles with the positive direction of co-ordinate axes is

Options

A. 3
B. 1
C. `sqrt(3)`
D. `(3)/(sqrt(2))`

Answer

To solve the problem, we need to find the sum of the direction cosines of a line that makes equal angles with the positive direction of the coordinate axes.

### Step-by-Step Solution:

1. **Understanding Direction Cosines**:
- Let the direction cosines of the line be denoted as \( l, m, n \).
- The direction cosines are defined as \( l = \cos \alpha \), \( m = \cos \beta \), and \( n = \cos \gamma \), where \( \alpha, \beta, \gamma \) are the angles that the line makes with the x, y, and z axes respectively.

2. **Equal Angles Condition**:
- According to the problem, the line makes equal angles with the coordinate axes. Therefore, we have:
\[ \alpha = \beta = \gamma \]
- This implies that:
\[ l = m = n = \cos \alpha \]

3. **Using the Direction Cosine Condition**:
- The sum of the squares of the direction cosines must equal 1:
\[ l^2 + m^2 + n^2 = 1 \]
- Substituting \( l, m, n \) with \( \cos \alpha \):
\[ \cos^2 \alpha + \cos^2 \alpha + \cos^2 \alpha = 1 \]
- This simplifies to:
\[ 3 \cos^2 \alpha = 1 \]

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

5. **Finding the Direction Cosines**:
- Therefore, the direction cosines can be:
\[ l = m = n = \frac{1}{\sqrt{3}} \quad \text{or} \quad l = m = n = -\frac{1}{\sqrt{3}} \]

6. **Calculating the Sum of Direction Cosines**:
- The sum of the direction cosines is:
\[ l + m + n = \cos \alpha + \cos \alpha + \cos \alpha = 3 \cos \alpha \]
- Substituting the values:
- For \( \cos \alpha = \frac{1}{\sqrt{3}} \):
\[ \text{Sum} = 3 \times \frac{1}{\sqrt{3}} = \sqrt{3} \]
- For \( \cos \alpha = -\frac{1}{\sqrt{3}} \):
\[ \text{Sum} = 3 \times -\frac{1}{\sqrt{3}} = -\sqrt{3} \]

7. **Conclusion**:
- The possible sums of the direction cosines are \( \sqrt{3} \) and \( -\sqrt{3} \). However, since the question asks for the sum of the direction cosines of a line that makes equal angles with the positive direction of the coordinate axes, the most appropriate answer is:
\[ \sqrt{3} \]