Find the direction cosines of that line whose direction ratios are as follows :
(i) `1,-2,2` , (ii) `2,6,3`
(iii) `3,1,-2`

Answer

To find the direction cosines of the lines given their direction ratios, we will use the formulas for direction cosines based on the direction ratios. The direction ratios are denoted by \( A, B, C \) and the direction cosines are denoted by \( L, M, N \). The relationships are as follows:

\[ L = \frac{A}{\sqrt{A^2 + B^2 + C^2}}, \quad M = \frac{B}{\sqrt{A^2 + B^2 + C^2}}, \quad N = \frac{C}{\sqrt{A^2 + B^2 + C^2}} \]

Now, we will solve for each set of direction ratios provided.

### Part (i): Direction Ratios \( \langle 1, -2, 2 \rangle \)

1. **Identify the direction ratios:**
- \( A = 1 \)
- \( B = -2 \)
- \( C = 2 \)

2. **Calculate \( A^2 + B^2 + C^2 \):**
\[ A^2 + B^2 + C^2 = 1^2 + (-2)^2 + 2^2 = 1 + 4 + 4 = 9 \]

3. **Calculate \( \sqrt{A^2 + B^2 + C^2} \):**
\[ \sqrt{9} = 3 \]

4. **Calculate direction cosines:**
\[ L = \frac{1}{3}, \quad M = \frac{-2}{3}, \quad N = \frac{2}{3} \]

Thus, the direction cosines for the first line are:
\[ \left( \frac{1}{3}, \frac{-2}{3}, \frac{2}{3} \right) \]

### Part (ii): Direction Ratios \( \langle 2, 6, 3 \rangle \)

1. **Identify the direction ratios:**
- \( A = 2 \)
- \( B = 6 \)
- \( C = 3 \)

2. **Calculate \( A^2 + B^2 + C^2 \):**
\[ A^2 + B^2 + C^2 = 2^2 + 6^2 + 3^2 = 4 + 36 + 9 = 49 \]

3. **Calculate \( \sqrt{A^2 + B^2 + C^2} \):**
\[ \sqrt{49} = 7 \]

4. **Calculate direction cosines:**
\[ L = \frac{2}{7}, \quad M = \frac{6}{7}, \quad N = \frac{3}{7} \]

Thus, the direction cosines for the second line are:
\[ \left( \frac{2}{7}, \frac{6}{7}, \frac{3}{7} \right) \]

### Part (iii): Direction Ratios \( \langle 3, 1, -2 \rangle \)

1. **Identify the direction ratios:**
- \( A = 3 \)
- \( B = 1 \)
- \( C = -2 \)

2. **Calculate \( A^2 + B^2 + C^2 \):**
\[ A^2 + B^2 + C^2 = 3^2 + 1^2 + (-2)^2 = 9 + 1 + 4 = 14 \]

3. **Calculate \( \sqrt{A^2 + B^2 + C^2} \):**
\[ \sqrt{14} \]

4. **Calculate direction cosines:**
\[ L = \frac{3}{\sqrt{14}}, \quad M = \frac{1}{\sqrt{14}}, \quad N = \frac{-2}{\sqrt{14}} \]

Thus, the direction cosines for the third line are:
\[ \left( \frac{3}{\sqrt{14}}, \frac{1}{\sqrt{14}}, \frac{-2}{\sqrt{14}} \right) \]

### Summary of Direction Cosines
1. For \( \langle 1, -2, 2 \rangle \): \( \left( \frac{1}{3}, \frac{-2}{3}, \frac{2}{3} \right) \)
2. For \( \langle 2, 6, 3 \rangle \): \( \left( \frac{2}{7}, \frac{6}{7}, \frac{3}{7} \right) \)
3. For \( \langle 3, 1, -2 \rangle \): \( \left( \frac{3}{\sqrt{14}}, \frac{1}{\sqrt{14}}, \frac{-2}{\sqrt{14}} \right) \)