The directions rations of two lines are 3,2,-6 and , 1,2,2 respectively . The acute
angle between these lines is

Options

A. `cos ^(-1) .((5)/(18))`
B. `cos^(-1) .((3)/(20))`
C. `cos^(-1).((5)/(21))`
D. `cos^(-1)((8)/(21))`

Explanation

To find the acute angle between two lines given their direction ratios, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Direction Ratios:**
The direction ratios of the two lines are given as:
- Line 1: \( P = (3, 2, -6) \)
- Line 2: \( Q = (1, 2, 2) \)

2. **Convert Direction Ratios to Vectors:**
We can express these direction ratios as vectors:
- \( \vec{P} = 3\hat{i} + 2\hat{j} - 6\hat{k} \)
- \( \vec{Q} = 1\hat{i} + 2\hat{j} + 2\hat{k} \)

3. **Calculate the Dot Product:**
The dot product \( \vec{P} \cdot \vec{Q} \) is calculated as follows:
\[ \vec{P} \cdot \vec{Q} = (3)(1) + (2)(2) + (-6)(2) = 3 + 4 - 12 = -5 \]

4. **Calculate the Magnitudes of the Vectors:**
- For \( \vec{P} \):
\[ |\vec{P}| = \sqrt{3^2 + 2^2 + (-6)^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7 \]
- For \( \vec{Q} \):
\[ |\vec{Q}| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

5. **Use the Cosine Formula:**
The cosine of the angle \( \theta \) between the two vectors is given by:
\[ \cos \theta = \frac{\vec{P} \cdot \vec{Q}}{|\vec{P}| |\vec{Q}|} \]
Substituting the values we found:
\[ \cos \theta = \frac{-5}{7 \times 3} = \frac{-5}{21} \]
Since we are looking for the acute angle, we take the absolute value:
\[ \cos \theta = \frac{5}{21} \]

6. **Calculate the Angle:**
To find \( \theta \), we take the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{5}{21}\right) \]

### Final Answer:
The acute angle \( \theta \) between the two lines is:
\[ \theta = \cos^{-1}\left(\frac{5}{21}\right) \]