The direction ratios of a line which is perpendicular to the two lines whose direction ratios are 3, -2, 1 and 2, 4, -2 is

Options

A. 0, 2, 1
B. 0, -1, 2
C. 0, 1, 2
D. 0, 1, -2

Answer

To find the direction ratios of a line that is perpendicular to two given lines with direction ratios \( (3, -2, 1) \) and \( (2, 4, -2) \), we can use the concept of the scalar product (dot product) of vectors.

### Step-by-step Solution:

1. **Identify the Direction Ratios**:
- Let the direction ratios of the first line \( L_1 \) be \( (A_1, B_1, C_1) = (3, -2, 1) \).
- Let the direction ratios of the second line \( L_2 \) be \( (A_2, B_2, C_2) = (2, 4, -2) \).

2. **Set Up the Perpendicular Condition**:
- For a line \( L_3 \) with direction ratios \( (A_3, B_3, C_3) \) to be perpendicular to \( L_1 \), the following equation must hold:
\[ A_1 A_3 + B_1 B_3 + C_1 C_3 = 0 \]
- Substituting the values:
\[ 3A_3 - 2B_3 + 1C_3 = 0 \quad \text{(1)} \]

3. **Set Up the Second Perpendicular Condition**:
- For \( L_3 \) to be perpendicular to \( L_2 \), we have:
\[ A_2 A_3 + B_2 B_3 + C_2 C_3 = 0 \]
- Substituting the values:
\[ 2A_3 + 4B_3 - 2C_3 = 0 \quad \text{(2)} \]

4. **Solve the System of Equations**:
- We now have two equations:
- \( 3A_3 - 2B_3 + C_3 = 0 \) (1)
- \( 2A_3 + 4B_3 - 2C_3 = 0 \) (2)

- From equation (1), we can express \( C_3 \):
\[ C_3 = -3A_3 + 2B_3 \quad \text{(3)} \]

- Substitute \( C_3 \) from equation (3) into equation (2):
\[ 2A_3 + 4B_3 - 2(-3A_3 + 2B_3) = 0 \]
\[ 2A_3 + 4B_3 + 6A_3 - 4B_3 = 0 \]
\[ 8A_3 = 0 \implies A_3 = 0 \]

5. **Substitute Back to Find \( B_3 \) and \( C_3 \)**:
- Substitute \( A_3 = 0 \) into equation (3):
\[ C_3 = -3(0) + 2B_3 = 2B_3 \]
- Now substitute \( A_3 = 0 \) into equation (1):
\[ 0 - 2B_3 + C_3 = 0 \implies C_3 = 2B_3 \]
- Let \( B_3 = 1 \) (we can choose any non-zero value for direction ratios):
\[ C_3 = 2(1) = 2 \]

6. **Final Direction Ratios**:
- Thus, the direction ratios of line \( L_3 \) are:
\[ (A_3, B_3, C_3) = (0, 1, 2) \]

### Conclusion:
The direction ratios of the line that is perpendicular to the two given lines are \( (0, 1, 2) \).