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

Options

A. `-8, 10, 11`
B. `8, -10,11`
C. `8, 10, -11`
D. `8, 10, 11`

Answer

To find the direction ratios of a line that is perpendicular to the two given lines with direction ratios \( \mathbf{a} = (3, -2, 4) \) and \( \mathbf{b} = (1, 3, -2) \), we can use the cross product of the two vectors represented by these direction ratios.

### Step-by-Step Solution:

1. **Identify the Direction Ratios**:
The direction ratios of the first line are \( \mathbf{a} = (3, -2, 4) \) and for the second line, they are \( \mathbf{b} = (1, 3, -2) \).

2. **Set Up the Cross Product**:
The direction ratios of the line perpendicular to both lines can be found using the cross product \( \mathbf{a} \times \mathbf{b} \).

3. **Calculate the Cross Product**:
The formula for the cross product of two vectors \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \) is given by:
\[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} \]
Substituting the values:
\[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 3 & -2 & 4 \\ 1 & 3 & -2 \end{vmatrix} \]

4. **Calculate the Determinants**:
Expanding the determinant:
\[ \mathbf{a} \times \mathbf{b} = \mathbf{i} \begin{vmatrix} -2 & 4 \\ 3 & -2 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 3 & 4 \\ 1 & -2 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 3 & -2 \\ 1 & 3 \end{vmatrix} \]

- For \( \mathbf{i} \):
\[ = (-2)(-2) - (4)(3) = 4 - 12 = -8 \]

- For \( \mathbf{j} \):
\[ = (3)(-2) - (4)(1) = -6 - 4 = -10 \quad \text{(note the negative sign in front)} \]
So, it becomes \( +10 \).

- For \( \mathbf{k} \):
\[ = (3)(3) - (-2)(1) = 9 + 2 = 11 \]

5. **Combine the Results**:
Therefore, the direction ratios of the line perpendicular to both lines are:
\[ \mathbf{a} \times \mathbf{b} = (-8, 10, 11) \]

### Final Answer:
The direction ratios of the required line are \( (-8, 10, 11) \).

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