if a line has direction ratio 2,-1,-2,determine its direction cosine

Explanation

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

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

### Step 2: Calculate the magnitude of the direction ratios
The magnitude (or length) of the direction ratios is calculated using the formula:
\[ \text{Magnitude} = \sqrt{a^2 + b^2 + c^2} \]
Substituting the values:
\[ \text{Magnitude} = \sqrt{2^2 + (-1)^2 + (-2)^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \]

### Step 3: Calculate the direction cosines
The direction cosines are calculated using the formulas:
\[ \cos \alpha = \frac{a}{\text{Magnitude}}, \quad \cos \beta = \frac{b}{\text{Magnitude}}, \quad \cos \gamma = \frac{c}{\text{Magnitude}} \]
Substituting the values:
\[ \cos \alpha = \frac{2}{3}, \quad \cos \beta = \frac{-1}{3}, \quad \cos \gamma = \frac{-2}{3} \]

### Step 4: Final Result
Thus, the direction cosines of the line are:
\[ \cos \alpha = \frac{2}{3}, \quad \cos \beta = \frac{-1}{3}, \quad \cos \gamma = \frac{-2}{3} \]

### Summary
The direction cosines are:
\[ \left( \frac{2}{3}, \frac{-1}{3}, \frac{-2}{3} \right) \]

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