If the direction cosines of a line are k,k,k then

Options

A. `k gt 0`
B. `0 lt k lt 1`
C. `k = 1`
D. `k=(1)/(sqrt3)or-(1)/(sqrt3)`

Answer

To solve the problem, we need to find the value of \( k \) given that the direction cosines of a line are \( k, k, k \).

### Step-by-Step Solution:

1. **Understanding Direction Cosines**:
The direction cosines of a line are defined as the cosines of the angles that the line makes with the coordinate axes. If the direction cosines are \( k, k, k \), we denote them as:
\[ L = k, \quad M = k, \quad N = k \]

2. **Using the Property of Direction Cosines**:
The sum of the squares of the direction cosines must equal 1. Therefore, we can write:
\[ L^2 + M^2 + N^2 = 1 \]
Substituting the values of \( L, M, \) and \( N \):
\[ k^2 + k^2 + k^2 = 1 \]

3. **Simplifying the Equation**:
This simplifies to:
\[ 3k^2 = 1 \]

4. **Solving for \( k^2 \)**:
To find \( k^2 \), we divide both sides by 3:
\[ k^2 = \frac{1}{3} \]

5. **Finding \( k \)**:
Taking the square root of both sides gives us:
\[ k = \pm \sqrt{\frac{1}{3}} = \pm \frac{1}{\sqrt{3}} \]

6. **Conclusion**:
Thus, the values of \( k \) are:
\[ k = \frac{1}{\sqrt{3}} \quad \text{and} \quad k = -\frac{1}{\sqrt{3}} \]

### Final Answer:
The values of \( k \) are \( \frac{1}{\sqrt{3}} \) and \( -\frac{1}{\sqrt{3}} \).

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