Question

A vector parallel to the line of intersection of the planes $\overrightarrow{r}·\left(3\hat{i}-\hat{j}+\hat{k}\right)=1\mathrm{and}\overrightarrow{r}·\left(\hat{i}+4\hat{j}-2\hat{k}\right)=2$ is
(a) $-2\hat{i}+7\hat{j}+13\hat{k}$

(b) $2\hat{i}+7\hat{j}-13\hat{k}$

(c) $-2\hat{i}-7\hat{j}+13\hat{k}$

(d) $2\hat{i}+7\hat{j}+13\hat{k}$

Answer 1

(b) $2\hat{i}+7\hat{j}-13\hat{k}$

$$\begin{gathered} \text{Let the required vector be}a\hat{i}+b\hat{j}+c\hat{k}...\left(1\right) \\ \text{Since the vector is parallel to the line of intersection of the given planes,} \\ 3a-b+c=0...\left(2\right) \\ a+4b-2c=0...\left(3\right) \\ \text{Solving (2) and (3), we get} \\ \frac{a}{-2}=\frac{b}{7}=\frac{c}{13} \\ \text{Substituting these values in (1), we get} \\ -2\hat{i}+7\hat{j}+13\hat{k},\text{which is the required vector.} \end{gathered}$$