The angle between two lines `(x)/(2)=(y)/(2)=(z)/(-1)and(x-1)/(1)=(y-1)/(2)=(z-1)/(2)` is

Options

A. `cos^(-1)((4)/(9))`
B. `cos^(-1)((1)/(3))`
C. `cos^(-1)((2)/(9))`
D. `cos^(-1)((5)/(9))`

Answer

To find the angle between the two lines given in the question, we will follow these steps:

### Step 1: Identify the direction ratios of the lines.
The first line is given by the equation:
\[ \frac{x}{2} = \frac{y}{2} = \frac{z}{-1} \]
From this, we can extract the direction ratios \( (a_1, b_1, c_1) \) as \( (2, 2, -1) \).

The second line is given by the equation:
\[ \frac{x-1}{1} = \frac{y-1}{2} = \frac{z-1}{2} \]
From this, we can extract the direction ratios \( (a_2, b_2, c_2) \) as \( (1, 2, 2) \).

### Step 2: Use the formula for the cosine of the angle between two lines.
The cosine of the angle \( \theta \) between two lines can be calculated using the formula:
\[ \cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}} \]

### Step 3: Calculate the dot product \( a_1 a_2 + b_1 b_2 + c_1 c_2 \).
Substituting the values:
\[ a_1 a_2 = 2 \cdot 1 = 2 \]
\[ b_1 b_2 = 2 \cdot 2 = 4 \]
\[ c_1 c_2 = -1 \cdot 2 = -2 \]
Now, summing these up:
\[ a_1 a_2 + b_1 b_2 + c_1 c_2 = 2 + 4 - 2 = 4 \]

### Step 4: Calculate the magnitudes of the direction ratios.
First line:
\[ \sqrt{a_1^2 + b_1^2 + c_1^2} = \sqrt{2^2 + 2^2 + (-1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]

Second line:
\[ \sqrt{a_2^2 + b_2^2 + c_2^2} = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

### Step 5: Substitute into the cosine formula.
Now, substituting the values into the cosine formula:
\[ \cos \theta = \frac{4}{3 \cdot 3} = \frac{4}{9} \]

### Step 6: Find the angle \( \theta \).
To find \( \theta \), we take the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{4}{9}\right) \]

### Conclusion
The angle between the two lines is \( \theta = \cos^{-1}\left(\frac{4}{9}\right) \).

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