Question

The direction ratios of the line which is perpendicular to the two lines ${\displaystyle \frac{x-7}{2}=\frac{y+17}{-3}=\frac{z-6}{1}\text{and}\frac{x+5}{1}=\frac{y+3}{2}=\frac{z-4}{-2}}$ are ______.

विकल्प

• 4, 5, 7

• 4, –5, 7

• 4, –5, –7

• –4, 5, 8

Answer 1

The direction ratios of the line which is perpendicular to the two lines ${\displaystyle \frac{x-7}{2}=\frac{y+17}{-3}=\frac{z-6}{1}\text{and}\frac{x+5}{1}=\frac{y+3}{2}=\frac{z-4}{-2}}$ are 4, 5, 7.

Explanation:

${\displaystyle \frac{x-7}{2}=\frac{y+17}{-3}=\frac{z-6}{1}}$

${\displaystyle \frac{x+5}{1}=\frac{y+3}{2}=\frac{z-4}{-2}}$

The direction ratios of the given lines are proportional to 2, -3, 1 and 1, 2, -2.

The given lines are parallel to the vectors →

${\displaystyle {\overrightarrow{b}}_1=2\hat{i}-3\hat{j}+\hat{k} \text{and} {\overrightarrow{b}}_2=\hat{i}+2\hat{j}-2\hat{k}}$

The vector perpendicular to the given two lines is →

${\displaystyle \overrightarrow{b}={\overrightarrow{b}}_1\times {\overrightarrow{b}}_2}$

= ${\displaystyle \left|\begin{array}{c}\hat{i} \hat{j} \hat{k}\\ 2-3 1\\ 1 2-2\end{array}\right|}$

= ${\displaystyle 4\hat{i}+5\hat{j}+7\hat{k}}$

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