# 19.  Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes $\overrightarrow{r}.\left ( \widehat{i}-\widehat{j}+2\widehat{k} \right )=5$ and $\overrightarrow{r}.\left ( 3\widehat{i}+\widehat{j}+\widehat{k} \right )=6$.

P Pankaj Sanodiya

Given

A point through which line passes

$\vec a=\hat i+2\hat j+3\hat k$

two plane

$\overrightarrow{r}.\left ( \widehat{i}-\widehat{j}+2\widehat{k} \right )=5$  And

$\overrightarrow{r}.\left ( 3\widehat{i}+\widehat{j}+\widehat{k} \right )=6$

it can be seen that normals of the planes are

$\vec n_1=\hat i-\hat j+2\hat k$

$\vec n_2=3\hat i+\hat j+\hat k$
since the line is parallel to both planes, its parallel vector will be perpendicular to normals of both planes.

So, a vector perpendicular to both these normal vector is

$\vec d=\vec n_1\times\vec n_2$

$\vec d=\begin{vmatrix} \hat i &\hat j & \hat k\\ 1 &-1 &2 \\ 3& 1 & 1 \end{vmatrix}=\hat i(-1-2)-\hat j(1-6)+\hat k(1+3)$

$\vec d=-3\hat i+5\hat j+4\hat k$

Now a line which passes through $\vec a$ and parallels to $\vec d$ is

$L=\vec a+\lambda\vec d$

So the required line is

$L=\vec a+\lambda\vec d$

$L=\hat i+2\hat j+3\hat k+\lambda(-3\hat i+5\hat j+4\hat k)$

$L=(1-3\lambda)\hat i+(2+5\lambda)\hat j+(3+4\lambda)\hat k$

Exams
Articles
Questions