Question

The direction cosines of a line which is perpendicular to lines whose direction ratios are \( 3, -2, 4 \) and \( 1, 3, -2 \) are

• A. \( \frac{-8}{\sqrt{285}}, \frac{-10}{\sqrt{285}}, \frac{11}{\sqrt{285}} \)
• B. \( \frac{-8}{\sqrt{285}}, \frac{10}{\sqrt{285}}, \frac{11}{\sqrt{285}} \)
• C. \( \frac{8}{\sqrt{285}}, \frac{10}{\sqrt{285}}, \frac{11}{\sqrt{285}} \)
• D. \( \frac{4}{\sqrt{297}}, \frac{5}{\sqrt{297}}, \frac{16}{\sqrt{297}} \)

Hint:

To find the direction cosines of a line perpendicular to two given lines, compute the cross product of their direction ratios.

Answer 1

The Correct Option is B

Step 1: Direction ratios of the given lines.
Let the direction ratios of the given lines be \( l_1 = 3, -2, 4 \) and \( l_2 = 1, 3, -2 \). The direction ratios of the line perpendicular to both of these lines can be found by taking the cross product of \( l_1 \) and \( l_2 \).

Step 2: Computing the cross product.
The cross product of \( l_1 = (3, -2, 4) \) and \( l_2 = (1, 3, -2) \) is: \[ l = \left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k}
3 & -2 & 4
1 & 3 & -2 \end{matrix} \right| \] After calculating the determinant, we find the direction ratios of the perpendicular line as \( \left( \frac{-8}{\sqrt{285}}, \frac{10}{\sqrt{285}}, \frac{11}{\sqrt{285}} \right) \).

Step 3: Conclusion.
Thus, the direction cosines of the perpendicular line are \( \frac{-8}{\sqrt{285}}, \frac{10}{\sqrt{285}}, \frac{11}{\sqrt{285}} \), which makes option (B) the correct answer.