If the direction ratios of a lines are proportional to `1,-3,2` then its direction cosines are

Options

A. `1/(sqrt(14)),(-3)/(sqrt(14)),2/(sqrt(14))`,
B. `1/(sqrt(14)),2/(sqrt(14)),3/(sqrt(14))`
C. `-1/(sqrt(14)),3/(sqrt(14)),2/(sqrt(14))`
D. `-1/(sqrt(14)),(-2)/(sqrt(14)),(-3)/(sqrt(14))`

Explanation

To find the direction cosines of a line whose direction ratios are proportional to \(1, -3, 2\), we can follow these steps:

### Step 1: Identify the Direction Ratios
The direction ratios given are proportional to \(1, -3, 2\). We can denote these as \(A = 1\), \(B = -3\), and \(C = 2\).

### Step 2: Calculate the Magnitude of the Direction Ratios
The magnitude of the direction ratios can be calculated using the formula:
\[ \sqrt{A^2 + B^2 + C^2} \]
Substituting the values:
\[ \sqrt{1^2 + (-3)^2 + 2^2} = \sqrt{1 + 9 + 4} = \sqrt{14} \]

### Step 3: Calculate the Direction Cosines
The direction cosines \(l\), \(m\), and \(n\) can be calculated using the formulas:
\[ l = \frac{A}{\sqrt{A^2 + B^2 + C^2}}, \quad m = \frac{B}{\sqrt{A^2 + B^2 + C^2}}, \quad n = \frac{C}{\sqrt{A^2 + B^2 + C^2}} \]
Substituting the values we have:
\[ l = \frac{1}{\sqrt{14}}, \quad m = \frac{-3}{\sqrt{14}}, \quad n = \frac{2}{\sqrt{14}} \]

### Step 4: Write the Direction Cosines
Thus, the direction cosines are:
\[ l = \frac{1}{\sqrt{14}}, \quad m = \frac{-3}{\sqrt{14}}, \quad n = \frac{2}{\sqrt{14}} \]

### Final Answer
The direction cosines of the line are:
\[ \left( \frac{1}{\sqrt{14}}, \frac{-3}{\sqrt{14}}, \frac{2}{\sqrt{14}} \right) \]

---