Question

A line $L_1$ passing through a point with position vector $\vec{p}=\hat{i}+2 \hat{j}+3 \hat{k}$ and parallel to $\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}$ . Another line $L _2$ passing through a point with position vector $\vec{q}=2 \hat{i}+3 \hat{j}+\hat{k}$ and parallel to $\vec{b}=3 \hat{i}+\hat{j}+2 \hat{k}$.
Equation of plane equidistant from lines $L_1$ and $L_2$ is

• A. $\hat{r} \cdot(\hat{i}-7 \hat{j}-5 \hat{k})=3$
• B. $\hat{r} \cdot(\hat{i}+7 \hat{j}+5 \hat{k})=3$
• C. $\hat{r} \cdot(\hat{i}-7 \hat{j}-5 \hat{k})=9$
• D. $\hat{r} \cdot(\hat{i}+7 \hat{j}-5 \hat{k})=9$

Answer 1

Answer: (D)

Line $L_1$ is parallel to $\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}$ Line $L_2$ is parallel to $\vec{b}=3 \hat{i}+\hat{j}+2 \hat{k}$
$\Rightarrow$ normal to the plane perpendicular to line $L_1$ and $L_2$ is $\vec{a} \times \vec{b}=(\hat{i}+7 \hat{j}-5 \hat{k})$ and plane passes through the point with positive vector
$=\frac{3}{2} \hat{i}+\frac{5}{2} \hat{j}+2 \hat{k}$
equation of plane is
$\hat{r}(\hat{i}+7 \hat{j}-5 \hat{k})=9.$