Find the angle between the following pairs of lines :
(i) `vec(r) = 3 hat(i) + 2 hat(j) - 4 hat(k) + lambda (hat(i) + 2 hat(j) + 2 hat(k) )`.
`vec(r) = 5 hat(j) - 2 hat(k) + mu ( 3 hat(i) + 2 hat(j) + 6 hat(k)) `
(ii) `vec(r) = 3 hat(i) + hat(j) - 2 hat(k) + lambda (hat(i) - hat(j) - 2 hat(k) )`.
`vec(r) =(2 hat(i) - hat(j) - 56 hat(k)) + mu ( 3 hat(i) - 5 hat(j) - 4 hat(k)) `
(iii) `(x-2)/(2) = (y - 1)/(5) = (z + 3)/(-3) `
and `(x + 2)/(-1) = (y - 4)/(8) = (z - 5)/(4)`
(iv) `(x - 4)/(3) = (y + 1)/(4) = (z - 6)/(5) and (x-5)/(1) = (2y +5)/(-2) = (z - 3)/(1)`
(v) `(5 -x)/(3) = (y + 3)/(-4) , z = 7 and x = (1-y)/(2) = (z - 6)/(2)`
(vi)` (x + 3)/(3) = (y - 1)/(5) = (z + 3)/(4) and (x + 1)/(1) = (y - 4)/(1) = (z-5)/(2) `.

Answer

To find the angle between the given pairs of lines, we will use the formula for the angle θ between two vectors **a** and **b**:

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

Where:
- \(\mathbf{a} \cdot \mathbf{b}\) is the dot product of vectors **a** and **b**.
- \(|\mathbf{a}|\) and \(|\mathbf{b}|\) are the magnitudes of vectors **a** and **b** respectively.

### (i)
Given lines:
1. \(\vec{r_1} = 3\hat{i} + 2\hat{j} - 4\hat{k} + \lambda (\hat{i} + 2\hat{j} + 2\hat{k})\)
2. \(\vec{r_2} = 5\hat{j} - 2\hat{k} + \mu (3\hat{i} + 2\hat{j} + 6\hat{k})\)

**Step 1:** Identify direction vectors.
- For the first line: \(\mathbf{a} = \hat{i} + 2\hat{j} + 2\hat{k}\)
- For the second line: \(\mathbf{b} = 3\hat{i} + 2\hat{j} + 6\hat{k}\)

**Step 2:** Calculate the dot product \(\mathbf{a} \cdot \mathbf{b}\).
\[ \mathbf{a} \cdot \mathbf{b} = (1)(3) + (2)(2) + (2)(6) = 3 + 4 + 12 = 19 \]

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

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

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

### (ii)
Given lines:
1. \(\vec{r_1} = 3\hat{i} + \hat{j} - 2\hat{k} + \lambda (\hat{i} - \hat{j} - 2\hat{k})\)
2. \(\vec{r_2} = (2\hat{i} - \hat{j} - 56\hat{k}) + \mu (3\hat{i} - 5\hat{j} - 4\hat{k})\)

**Step 1:** Identify direction vectors.
- For the first line: \(\mathbf{a} = \hat{i} - \hat{j} - 2\hat{k}\)
- For the second line: \(\mathbf{b} = 3\hat{i} - 5\hat{j} - 4\hat{k}\)

**Step 2:** Calculate the dot product \(\mathbf{a} \cdot \mathbf{b}\).
\[ \mathbf{a} \cdot \mathbf{b} = (1)(3) + (-1)(-5) + (-2)(-4) = 3 + 5 + 8 = 16 \]

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

**Step 4:** Substitute into the cosine formula.
\[ \cos \theta = \frac{16}{\sqrt{6} \cdot \sqrt{50}} = \frac{16}{\sqrt{300}} = \frac{16}{10\sqrt{3}} = \frac{8}{5\sqrt{3}} \]

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

### (iii)
Given lines:
1. \(\frac{x-2}{2} = \frac{y-1}{5} = \frac{z+3}{-3}\)
2. \(\frac{x+2}{-1} = \frac{y-4}{8} = \frac{z-5}{4}\)

**Step 1:** Identify direction vectors.
- For the first line: \(\mathbf{a} = 2\hat{i} + 5\hat{j} - 3\hat{k}\)
- For the second line: \(\mathbf{b} = -1\hat{i} + 8\hat{j} + 4\hat{k}\)

**Step 2:** Calculate the dot product \(\mathbf{a} \cdot \mathbf{b}\).
\[ \mathbf{a} \cdot \mathbf{b} = (2)(-1) + (5)(8) + (-3)(4) = -2 + 40 - 12 = 26 \]

**Step 3:** Calculate the magnitudes \(|\mathbf{a}|\) and \(|\mathbf{b}|\).
\[ |\mathbf{a}| = \sqrt{2^2 + 5^2 + (-3)^2} = \sqrt{4 + 25 + 9} = \sqrt{38} \]
\[ |\mathbf{b}| = \sqrt{(-1)^2 + 8^2 + 4^2} = \sqrt{1 + 64 + 16} = \sqrt{81} = 9 \]

**Step 4:** Substitute into the cosine formula.
\[ \cos \theta = \frac{26}{\sqrt{38} \cdot 9} \]

**Step 5:** Find the angle \(\theta\).
\[ \theta = \cos^{-1}\left(\frac{26}{9\sqrt{38}}\right) \]

### (iv)
Given lines:
1. \(\frac{x-4}{3} = \frac{y+1}{4} = \frac{z-6}{5}\)
2. \(\frac{x-5}{1} = \frac{2y+5}{-2} = \frac{z-3}{1}\)

**Step 1:** Identify direction vectors.
- For the first line: \(\mathbf{a} = 3\hat{i} + 4\hat{j} + 5\hat{k}\)
- For the second line: \(\mathbf{b} = \hat{i} - 2\hat{j} + \hat{k}\)

**Step 2:** Calculate the dot product \(\mathbf{a} \cdot \mathbf{b}\).
\[ \mathbf{a} \cdot \mathbf{b} = (3)(1) + (4)(-2) + (5)(1) = 3 - 8 + 5 = 0 \]

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

**Step 4:** Substitute into the cosine formula.
\[ \cos \theta = \frac{0}{\sqrt{50} \cdot \sqrt{6}} = 0 \]

**Step 5:** Find the angle \(\theta\).
\[ \theta = \cos^{-1}(0) = 90^\circ \]

### (v)
Given lines:
1. \(\frac{5-x}{3} = \frac{y+3}{-4}, z = 7\)
2. \(x = \frac{1-y}{2} = \frac{z-6}{2}\)

**Step 1:** Identify direction vectors.
- For the first line: \(\mathbf{a} = -3\hat{i} - 4\hat{j} + 0\hat{k}\)
- For the second line: \(\mathbf{b} = 2\hat{i} - 2\hat{j} + 2\hat{k}\)

**Step 2:** Calculate the dot product \(\mathbf{a} \cdot \mathbf{b}\).
\[ \mathbf{a} \cdot \mathbf{b} = (-3)(2) + (-4)(-2) + (0)(2) = -6 + 8 + 0 = 2 \]

**Step 3:** Calculate the magnitudes \(|\mathbf{a}|\) and \(|\mathbf{b}|\).
\[ |\mathbf{a}| = \sqrt{(-3)^2 + (-4)^2 + 0^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
\[ |\mathbf{b}| = \sqrt{2^2 + (-2)^2 + 2^2} = \sqrt{4 + 4 + 4} = \sqrt{12} = 2\sqrt{3} \]

**Step 4:** Substitute into the cosine formula.
\[ \cos \theta = \frac{2}{5 \cdot 2\sqrt{3}} = \frac{1}{5\sqrt{3}} \]

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

### (vi)
Given lines:
1. \(\frac{x+3}{3} = \frac{y-1}{5} = \frac{z+3}{4}\)
2. \(\frac{x+1}{1} = \frac{y-4}{1} = \frac{z-5}{2}\)

**Step 1:** Identify direction vectors.
- For the first line: \(\mathbf{a} = 3\hat{i} + 5\hat{j} + 4\hat{k}\)
- For the second line: \(\mathbf{b} = \hat{i} + \hat{j} + 2\hat{k}\)

**Step 2:** Calculate the dot product \(\mathbf{a} \cdot \mathbf{b}\).
\[ \mathbf{a} \cdot \mathbf{b} = (3)(1) + (5)(1) + (4)(2) = 3 + 5 + 8 = 16 \]

**Step 3:** Calculate the magnitudes \(|\mathbf{a}|\) and \(|\mathbf{b}|\).
\[ |\mathbf{a}| = \sqrt{3^2 + 5^2 + 4^2} = \sqrt{9 + 25 + 16} = \sqrt{50} \]
\[ |\mathbf{b}| = \sqrt{1^2 + 1^2 + 2^2} = \sqrt{1 + 1 + 4} = \sqrt{6} \]

**Step 4:** Substitute into the cosine formula.
\[ \cos \theta = \frac{16}{\sqrt{50} \cdot \sqrt{6}} = \frac{16}{\sqrt{300}} = \frac{16}{10\sqrt{3}} = \frac{8}{5\sqrt{3}} \]

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