Question

Find the vector and cartesian eqns of a plane containing the two lines.
$\overrightarrow{r}=(2\hat{i}+\hat{j}-3\hat{k})+\lambda (\hat{i}+2\hat{j}+5\hat{k})$ and $\overrightarrow{r}=(3\hat{i}+3\hat{j}+2\hat{k})+\mu (3\hat{i}-2\hat{j}+5\hat{k})$

Answer 1

$\to$ Normal vector of plane

would be perpendicular to

both the lines say $l_1$ & $l_2$

$\Rightarrow$ parallel to $\overrightarrow{l_1}\times \overrightarrow{l_2}$

$\overrightarrow{l_1}\times \overrightarrow{l_2}$ $=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 2& 5\\ 3& -2& 5\end{array}\right|$

$=20\hat{i}+10\hat{j}-6\hat{k}$

and the plane passes through

$(2,1,-3)$(line $l_1)$

$\Rightarrow$ its equation : $(\overrightarrow{r}-\overrightarrow{a})\overrightarrow{n}=0$

$\Rightarrow \overrightarrow{r}.\overrightarrow{n}=\overrightarrow{a}.\overrightarrow{n}$

$\Rightarrow \overrightarrow{r}(20\hat{i}+10\hat{j}-8\hat{k})=40+10+24$

$=74$

$\Rightarrow 20x+10y-8z=74$