The direction cosines of a line whose direction ratios are 1, -2, 3 are

Options

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

Answer

To find the direction cosines of a line given its direction ratios, we can follow these steps:

### Step 1: Identify the Direction Ratios
The direction ratios given are \( a = 1 \), \( b = -2 \), and \( c = 3 \).

### Step 2: Calculate the Magnitude of the Direction Ratios
To find the direction cosines, we first need to calculate the magnitude of the direction ratios. The magnitude \( R \) is given by the formula:
\[ R = \sqrt{a^2 + b^2 + c^2} \]
Substituting the values:
\[ R = \sqrt{1^2 + (-2)^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \]

### Step 3: Calculate the Direction Cosines
The direction cosines \( l \), \( m \), and \( n \) are calculated using the formulas:
\[ l = \frac{a}{R}, \quad m = \frac{b}{R}, \quad n = \frac{c}{R} \]

Substituting the values:
- For \( l \):
\[ l = \frac{1}{\sqrt{14}} \]

- For \( m \):
\[ m = \frac{-2}{\sqrt{14}} \]

- For \( n \):
\[ n = \frac{3}{\sqrt{14}} \]

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

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

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