Question

The direction ratios of the line perpendicular to the lines having direction ratios $2,3,1$ and $1,2,1$ are

• A. $-2,\,1,\,1$
• B. $1,\,1,\,1$
• C. $1,\,-1,\,1$
• D. $2,\,2,\,-2$

Hint:

A line perpendicular to two given lines has direction ratios equal to the cross product of their direction vectors.

Answer 1

The Correct Option is C

Step 1: Writing direction vectors of the given lines.
\[ \vec d_1=\langle2,3,1\rangle,\quad \vec d_2=\langle1,2,1\rangle \]
Step 2: Finding a vector perpendicular to both lines.
The required direction ratios are proportional to \[ \vec d_1\times\vec d_2 \] \[ =\begin{vmatrix} \hat i & \hat j & \hat k\\ 2 & 3 & 1\\ 1 & 2 & 1 \end{vmatrix} =\langle1,-1,1\rangle \]
Step 3: Conclusion.
The direction ratios are $1,\,-1,\,1$.