The angle between the lines whose direction-ratios are :
` lt 2, 1 , 2 gt and lt 4, 8, 1 gt ` is :

Options

A. `cos^(-1) ((3)/(2))`
B. `cos^(-1) ((2)/(3))`
C. `cos^(-1) ((10)/(3))`
D. `cos^(-1) ((1)/(3))`

Answer

To find the angle between the two lines with given direction ratios, we can use the formula for the cosine of the angle θ between two vectors:

\[ \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|} \]

where \(\mathbf{a}\) and \(\mathbf{b}\) are the direction ratios of the two lines.

### Step-by-Step Solution:

1. **Identify the Direction Ratios**:
The direction ratios of the first line \(L_1\) are given as \(\mathbf{a} = \langle 2, 1, 2 \rangle\) and for the second line \(L_2\) as \(\mathbf{b} = \langle 4, 8, 1 \rangle\).

2. **Calculate the Dot Product**:
The dot product \(\mathbf{a} \cdot \mathbf{b}\) is calculated as follows:
\[ \mathbf{a} \cdot \mathbf{b} = (2)(4) + (1)(8) + (2)(1) = 8 + 8 + 2 = 18 \]

3. **Calculate the Magnitudes of the Vectors**:
The magnitude of vector \(\mathbf{a}\) is:
\[ |\mathbf{a}| = \sqrt{2^2 + 1^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \]
The magnitude of vector \(\mathbf{b}\) is:
\[ |\mathbf{b}| = \sqrt{4^2 + 8^2 + 1^2} = \sqrt{16 + 64 + 1} = \sqrt{81} = 9 \]

4. **Substitute into the Cosine Formula**:
Now substitute the values into the cosine formula:
\[ \cos \theta = \frac{18}{3 \times 9} = \frac{18}{27} = \frac{2}{3} \]

5. **Find the Angle θ**:
To find the angle θ, we take the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{2}{3}\right) \]

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