Find the acute angle between the lines whose direction-ratios are :
` lt 2, 3, 6 gt and lt 1,2,-2, gt ` .

Answer

To find the acute angle between the lines with direction ratios \( \mathbf{l_1} = \langle 2, 3, 6 \rangle \) and \( \mathbf{l_2} = \langle 1, 2, -2 \rangle \), we can use the formula for the cosine of the angle \( \theta \) between two vectors:

\[ \cos \theta = \frac{\mathbf{l_1} \cdot \mathbf{l_2}}{|\mathbf{l_1}| |\mathbf{l_2}|} \]

### Step 1: Calculate the dot product \( \mathbf{l_1} \cdot \mathbf{l_2} \)

The dot product is calculated as follows:

\[ \mathbf{l_1} \cdot \mathbf{l_2} = 2 \cdot 1 + 3 \cdot 2 + 6 \cdot (-2) \]

Calculating each term:

\[ = 2 + 6 - 12 = -4 \]

### Step 2: Calculate the magnitudes \( |\mathbf{l_1}| \) and \( |\mathbf{l_2}| \)

The magnitude of \( \mathbf{l_1} \) is:

\[ |\mathbf{l_1}| = \sqrt{2^2 + 3^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]

The magnitude of \( \mathbf{l_2} \) is:

\[ |\mathbf{l_2}| = \sqrt{1^2 + 2^2 + (-2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

### Step 3: Substitute into the cosine formula

Now substituting into the cosine formula:

\[ \cos \theta = \frac{-4}{7 \cdot 3} = \frac{-4}{21} \]

### Step 4: Find the angle \( \theta \)

To find \( \theta \), we take the inverse cosine:

\[ \theta = \cos^{-1}\left(\frac{-4}{21}\right) \]

Since we are looking for the acute angle, we take the absolute value of the cosine:

\[ \theta = \cos^{-1}\left(\left|\frac{-4}{21}\right|\right) = \cos^{-1}\left(\frac{4}{21}\right) \]

### Final Answer

The acute angle \( \theta \) between the lines is:

\[ \theta = \cos^{-1}\left(\frac{4}{21}\right) \]