Question

The vector equation of the line of intersection of the planes $\overrightarrow{r}.(\hat{i}+2\hat{j}+3\hat{k})=0$ and $\overrightarrow{r}.(3\hat{i}+2\hat{j}+\hat{k})=0$ is

• A. $\overrightarrow{r}=\lambda (\hat{i}+2\hat{j}+\hat{k})$
• B. $\overrightarrow{r}=\lambda (\hat{i}-2\hat{j}+\hat{k})$
• C. $\overrightarrow{r}=\lambda (\hat{i}+2\hat{j}-3\hat{k})$
• D. None of these

Answer 1

The correct option is B $\overrightarrow{r}=\lambda (\hat{i}-2\hat{j}+\hat{k})$

The line of intersection of the planes $\overrightarrow{r}.(\hat{i}+2\hat{j}+3\hat{k})=0$ and $\overrightarrow{r}.(\widehat{3i}+2\hat{j}+\hat{k})=0$ is parallel to the vector $(\hat{i}+2\hat{j}+3\hat{k})\times (3\hat{i}+2\hat{j}+\hat{k})=-4\hat{i}+8\hat{j}-4\hat{k}$.
Since both the planes pass through the origin, therefore their line of intersection will also pass through the origin.
Thus, the required line passes through the origin and is parallel to the vector $-4\hat{i}+8\hat{j}-4\hat{k}$.
Hence, its equation is $\overrightarrow{r}=\overrightarrow{0}+{\lambda }^{\prime }(-4\hat{i}+8\hat{j}-4\hat{k})\Rightarrow \overrightarrow{r}=\lambda (\hat{i}-2\hat{j}+\hat{k})$
where $\lambda =4{\lambda }^{\prime }$