Find the angle between the following pairs of lines :
(i)`vec(r) = 2 hati - 5 hatj + hatk + lambda (3 hati + 2 hatj + 6 hatk )` and
`vec(r) = 7 hati - 6 hatk + mu (hati + 2 hatj + 2 hatk)`
(ii) ` vec(r) = 3 hati + hatj - 2 hatk + lambda (hati - hatj - 2 hatk )` and
`vec(r) = 2 hati - hatj - 56 hatk + mu (3 hati - 5 hatj - 4 hatk)`.

Answer

To find the angle between the given pairs of lines, we will use the formula for the cosine of the angle between two lines represented in vector form. The formula is:

\[ \cos \theta = \frac{\vec{b_1} \cdot \vec{b_2}}{|\vec{b_1}| |\vec{b_2}|} \]

where \(\vec{b_1}\) and \(\vec{b_2}\) are the direction ratios of the two lines, and \(\theta\) is the angle between them.

### Part (i)

1. **Identify the direction vectors:**
- For the first line \(L_1\):
\[ \vec{r} = 2\hat{i} - 5\hat{j} + \hat{k} + \lambda(3\hat{i} + 2\hat{j} + 6\hat{k}) \]
Here, \(\vec{b_1} = 3\hat{i} + 2\hat{j} + 6\hat{k}\).

- For the second line \(L_2\):
\[ \vec{r} = 7\hat{i} - 6\hat{k} + \mu(\hat{i} + 2\hat{j} + 2\hat{k}) \]
Here, \(\vec{b_2} = \hat{i} + 2\hat{j} + 2\hat{k}\).

2. **Calculate the dot product \(\vec{b_1} \cdot \vec{b_2}\):**
\[ \vec{b_1} \cdot \vec{b_2} = (3\hat{i} + 2\hat{j} + 6\hat{k}) \cdot (\hat{i} + 2\hat{j} + 2\hat{k}) \]
\[ = 3 \cdot 1 + 2 \cdot 2 + 6 \cdot 2 = 3 + 4 + 12 = 19 \]

3. **Calculate the magnitudes \(|\vec{b_1}|\) and \(|\vec{b_2}|\):**
\[ |\vec{b_1}| = \sqrt{3^2 + 2^2 + 6^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7 \]
\[ |\vec{b_2}| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

4. **Substitute into the cosine formula:**
\[ \cos \theta = \frac{19}{7 \cdot 3} = \frac{19}{21} \]

5. **Find the angle \(\theta\):**
\[ \theta = \cos^{-1}\left(\frac{19}{21}\right) \]

### Part (ii)

1. **Identify the direction vectors:**
- For the first line \(L_1\):
\[ \vec{r} = 3\hat{i} + \hat{j} - 2\hat{k} + \lambda(\hat{i} - \hat{j} - 2\hat{k}) \]
Here, \(\vec{b_1} = \hat{i} - \hat{j} - 2\hat{k}\).

- For the second line \(L_2\):
\[ \vec{r} = 2\hat{i} - \hat{j} - 56\hat{k} + \mu(3\hat{i} - 5\hat{j} - 4\hat{k}) \]
Here, \(\vec{b_2} = 3\hat{i} - 5\hat{j} - 4\hat{k}\).

2. **Calculate the dot product \(\vec{b_1} \cdot \vec{b_2}\):**
\[ \vec{b_1} \cdot \vec{b_2} = (\hat{i} - \hat{j} - 2\hat{k}) \cdot (3\hat{i} - 5\hat{j} - 4\hat{k}) \]
\[ = 1 \cdot 3 + (-1) \cdot (-5) + (-2) \cdot (-4) = 3 + 5 + 8 = 16 \]

3. **Calculate the magnitudes \(|\vec{b_1}|\) and \(|\vec{b_2}|\):**
\[ |\vec{b_1}| = \sqrt{1^2 + (-1)^2 + (-2)^2} = \sqrt{1 + 1 + 4} = \sqrt{6} \]
\[ |\vec{b_2}| = \sqrt{3^2 + (-5)^2 + (-4)^2} = \sqrt{9 + 25 + 16} = \sqrt{50} = 5\sqrt{2} \]

4. **Substitute into the cosine formula:**
\[ \cos \alpha = \frac{16}{\sqrt{6} \cdot 5\sqrt{2}} = \frac{16}{5\sqrt{12}} = \frac{16}{10\sqrt{3}} = \frac{8}{5\sqrt{3}} \]

5. **Find the angle \(\alpha\):**
\[ \alpha = \cos^{-1}\left(\frac{8}{5\sqrt{3}}\right) \]