Find the angle between those lines whose direction ratios are as follows :
(i) `(2,3,6)` and `(1,2,2)`
(ii) `(4,-3,5)` and `(3,4,5)`
(iii) `(1,2,1)` and `(4,-3,2)`

Answer

To find the angle between two lines given their direction ratios, we can use the formula:

\[ \cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \cdot \sqrt{a_2^2 + b_2^2 + c_2^2}} \]

where \((a_1, b_1, c_1)\) and \((a_2, b_2, c_2)\) are the direction ratios of the two lines, and \(\theta\) is the angle between them.

### Part (i): Direction Ratios (2, 3, 6) and (1, 2, 2)

1. **Identify the direction ratios**:
- Line 1: \( (a_1, b_1, c_1) = (2, 3, 6) \)
- Line 2: \( (a_2, b_2, c_2) = (1, 2, 2) \)

2. **Calculate the dot product**:
\[ a_1 a_2 + b_1 b_2 + c_1 c_2 = 2 \cdot 1 + 3 \cdot 2 + 6 \cdot 2 = 2 + 6 + 12 = 20 \]

3. **Calculate the magnitudes**:
\[ \sqrt{a_1^2 + b_1^2 + c_1^2} = \sqrt{2^2 + 3^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]
\[ \sqrt{a_2^2 + b_2^2 + c_2^2} = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

4. **Substitute into the formula**:
\[ \cos \theta = \frac{20}{7 \cdot 3} = \frac{20}{21} \]

5. **Find \(\theta\)**:
\[ \theta = \cos^{-1}\left(\frac{20}{21}\right) \]

### Part (ii): Direction Ratios (4, -3, 5) and (3, 4, 5)

1. **Identify the direction ratios**:
- Line 1: \( (a_1, b_1, c_1) = (4, -3, 5) \)
- Line 2: \( (a_2, b_2, c_2) = (3, 4, 5) \)

2. **Calculate the dot product**:
\[ a_1 a_2 + b_1 b_2 + c_1 c_2 = 4 \cdot 3 + (-3) \cdot 4 + 5 \cdot 5 = 12 - 12 + 25 = 25 \]

3. **Calculate the magnitudes**:
\[ \sqrt{a_1^2 + b_1^2 + c_1^2} = \sqrt{4^2 + (-3)^2 + 5^2} = \sqrt{16 + 9 + 25} = \sqrt{50} \]
\[ \sqrt{a_2^2 + b_2^2 + c_2^2} = \sqrt{3^2 + 4^2 + 5^2} = \sqrt{9 + 16 + 25} = \sqrt{50} \]

4. **Substitute into the formula**:
\[ \cos \theta = \frac{25}{\sqrt{50} \cdot \sqrt{50}} = \frac{25}{50} = \frac{1}{2} \]

5. **Find \(\theta\)**:
\[ \theta = \cos^{-1}\left(\frac{1}{2}\right) = 60^\circ \]

### Part (iii): Direction Ratios (1, 2, 1) and (4, -3, 2)

1. **Identify the direction ratios**:
- Line 1: \( (a_1, b_1, c_1) = (1, 2, 1) \)
- Line 2: \( (a_2, b_2, c_2) = (4, -3, 2) \)

2. **Calculate the dot product**:
\[ a_1 a_2 + b_1 b_2 + c_1 c_2 = 1 \cdot 4 + 2 \cdot (-3) + 1 \cdot 2 = 4 - 6 + 2 = 0 \]

3. **Calculate the magnitudes**:
\[ \sqrt{a_1^2 + b_1^2 + c_1^2} = \sqrt{1^2 + 2^2 + 1^2} = \sqrt{1 + 4 + 1} = \sqrt{6} \]
\[ \sqrt{a_2^2 + b_2^2 + c_2^2} = \sqrt{4^2 + (-3)^2 + 2^2} = \sqrt{16 + 9 + 4} = \sqrt{29} \]

4. **Substitute into the formula**:
\[ \cos \theta = \frac{0}{\sqrt{6} \cdot \sqrt{29}} = 0 \]

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

### Summary of Angles:
- (i) \( \theta = \cos^{-1}\left(\frac{20}{21}\right) \)
- (ii) \( \theta = 60^\circ \)
- (iii) \( \theta = 90^\circ \)