Question

The direction ratios of the line perpendicular to the lines \(\frac{x-7}{-6}= \frac{y+17}{4}= \frac{z-6}{2} \space and \space \frac {x+5}{6}=\frac{y+3}{3}=\frac{z-4}{-6}\) are proportional to:

• A. 7, 4, 5
• B. 7, 5, 4
• C. 5, 7, 4
• D. 5, 4, 7

Answer 1

The Correct Option is D

To find the direction ratios of a line perpendicular to the given lines, we first determine the direction ratios of each line. The line represented by \(\frac{x-7}{-6}=\frac{y+17}{4}=\frac{z-6}{2}\) has direction ratios \((-6,4,2)\). The line represented by \(\frac{x+5}{6}=\frac{y+3}{3}=\frac{z-4}{-6}\) has direction ratios \((6,3,-6)\). The direction ratios of a line perpendicular to both given lines can be found by taking the cross product of their direction ratios:

Given two vectors a \((-6,4,2)\) and b \((6,3,-6)\), the cross product c = a × b is computed as:

\[ \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -6 & 4 & 2 \\ 6 & 3 & -6 \end{vmatrix} \]

Expanding the determinant:

\[\mathbf{i}(4 \cdot -6 - 2 \cdot 3) - \mathbf{j}(-6 \cdot -6 - 2 \cdot 6) + \mathbf{k}(-6 \cdot 3 - 4 \cdot 6)\]

\[= \mathbf{i}(-24 - 6) - \mathbf{j}(36 - 12) + \mathbf{k}(-18 - 24)\]

\[= \mathbf{i}(-30) - \mathbf{j}(24) + \mathbf{k}(-42)\]

\(\Rightarrow\) Direction ratios of line perpendicular to both = \((-30, -24, -42)\).

These can be simplified by dividing each component by \(-6\) (greatest common divisor):

\((-30, -24, -42) \div -6 = (5, 4, 7)\)

Thus, the direction ratios of the line perpendicular to both given lines are proportional to 5, 4, 7.