The angle between a line with direction ratios proportional to `2, 2, 1` and a line joining `(3, 1, 4) and (7, 2, 12)` is

Options

A. `cos^(-1)((2)/(3))`
B. `cos^(-1)((-2)/(3))`
C. `tan^(-1)((2)/(3))`
D. None of these

Answer

To find the angle between the line with direction ratios proportional to \(2, 2, 1\) and the line joining the points \((3, 1, 4)\) and \((7, 2, 12)\), we can follow these steps:

### Step 1: Determine the direction ratios of the line joining the two points
The direction ratios of the line joining the points \((3, 1, 4)\) and \((7, 2, 12)\) can be calculated as follows:
\[ \text{Direction ratios} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \]
Substituting the coordinates:
\[ = (7 - 3, 2 - 1, 12 - 4) = (4, 1, 8) \]

### Step 2: Identify the direction ratios of the first line
The direction ratios of the first line are given as proportional to \(2, 2, 1\). We can denote these as:
\[ (l_1, m_1, n_1) = (2, 2, 1) \]

### Step 3: Set the direction ratios of the second line
From Step 1, we have the direction ratios of the second line as:
\[ (l_2, m_2, n_2) = (4, 1, 8) \]

### Step 4: Use the formula for the cosine of the angle between two lines
The cosine of the angle \(\theta\) between two lines with direction ratios \((l_1, m_1, n_1)\) and \((l_2, m_2, n_2)\) is given by:
\[ \cos \theta = \frac{l_1 l_2 + m_1 m_2 + n_1 n_2}{\sqrt{l_1^2 + m_1^2 + n_1^2} \cdot \sqrt{l_2^2 + m_2^2 + n_2^2}} \]

### Step 5: Substitute the values into the formula
Substituting the values we have:
\[ \cos \theta = \frac{(2 \cdot 4) + (2 \cdot 1) + (1 \cdot 8)}{\sqrt{2^2 + 2^2 + 1^2} \cdot \sqrt{4^2 + 1^2 + 8^2}} \]
Calculating the numerator:
\[ = \frac{8 + 2 + 8}{\sqrt{4 + 4 + 1} \cdot \sqrt{16 + 1 + 64}} = \frac{18}{\sqrt{9} \cdot \sqrt{81}} = \frac{18}{3 \cdot 9} = \frac{18}{27} = \frac{2}{3} \]

### Step 6: Find the angle \(\theta\)
Now we can find the angle \(\theta\) using:
\[ \theta = \cos^{-1}\left(\frac{2}{3}\right) \]

### Final Answer
Thus, the angle between the two lines is:
\[ \theta = \cos^{-1}\left(\frac{2}{3}\right) \]