Question

The equation of the plane containing the lines

$\overrightarrow{r}={\overrightarrow{a}}_1+\lambda {\overrightarrow{a}}_{2}and\overrightarrow{r}={\overrightarrow{a}}_2+\mu {\overrightarrow{a}}_1$ is

• A. $\left[\overrightarrow{r}{\overrightarrow{a}}_1{\overrightarrow{a}}_2\right]=0$
• B. $\left[\overrightarrow{r}{\overrightarrow{a}}_1{\overrightarrow{a}}_2\right]={\overrightarrow{a}}_1.{\overrightarrow{a}}_2$
• C. $\left[\overrightarrow{r}{\overrightarrow{a}}_1{\overrightarrow{a}}_2\right]=|{\overrightarrow{a}}_1||{\overrightarrow{a}}_2|$
• D. None of these

Answer 1

The correct option is A

$\left[\overrightarrow{r}{\overrightarrow{a}}_1{\overrightarrow{a}}_2\right]=0$

The required plane passes through a point having position vector ${\overrightarrow{a}}_1$ and is parallel to the vectors ${\overrightarrow{a}}_{1}and{\overrightarrow{a}}_2.$ If $\overrightarrow{r}$ is the position vector of any point on the plane, then $(\overrightarrow{r}-{\overrightarrow{a}}_1),{\overrightarrow{a}}_1,{\overrightarrow{a}}_2$ are coplanar.
Therefore $(\overrightarrow{r}-{\overrightarrow{a}}_1).({\overrightarrow{a}}_1\times {\overrightarrow{a}}_2)=0\Rightarrow \left[\overrightarrow{r}{\overrightarrow{a}}_1{\overrightarrow{a}}_2\right]=\left[{\overrightarrow{a}}_1{\overrightarrow{a}}_1{\overrightarrow{a}}_2\right]\Rightarrow \left[\overrightarrow{r}{\overrightarrow{a}}_1{\overrightarrow{a}}_2\right]=0(\because [{\overrightarrow{a}}_1{\overrightarrow{a}}_1{\overrightarrow{a}}_2]=0)$
Hence, the required plane is $\left[\overrightarrow{r}{\overrightarrow{a}}_1{\overrightarrow{a}}_2\right]=0$