एक रेखा की दिक्-कोसाइन ज्ञात कीजिए तो निर्देशांक्षो के साथ समान कोण बनाती है |

Options

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

Answer

**Step-by-Step Solution:**

1. **Understanding Direction Cosines**:
Direction cosines of a line are the cosines of the angles that the line makes with the coordinate axes. If a line makes angles α, β, and γ with the x, y, and z axes respectively, then the direction cosines are given by cos(α), cos(β), and cos(γ).

2. **Given Condition**:
The problem states that the line makes equal angles with the coordinate axes. This means that α = β = γ.

3. **Using the Property of Direction Cosines**:
Since the angles are equal, we can denote them as α. Therefore, we have:
- cos(α) = cos(β) = cos(γ)

4. **Relation of Direction Cosines**:
The sum of the squares of the direction cosines is equal to 1:
\[ \cos^2(α) + \cos^2(β) + \cos^2(γ) = 1 \]
Since cos(α) = cos(β) = cos(γ), we can denote this common value as 'l'. Thus, we can write:
\[ 3l^2 = 1 \]

5. **Solving for l**:
From the equation \(3l^2 = 1\), we can solve for l:
\[ l^2 = \frac{1}{3} \]
\[ l = \pm \frac{1}{\sqrt{3}} \]

6. **Conclusion**:
Therefore, the direction cosines of the line that makes equal angles with the coordinate axes are:
\[ \cos(α) = \cos(β) = \cos(γ) = \pm \frac{1}{\sqrt{3}} \]