Question

Find the vector equation of the following plane in cartesian form :
$\overrightarrow{r}=\hat{i}-\hat{j}+\lambda (\hat{i}+\hat{j}+\hat{k})+\mu (\hat{i}-2\hat{j}+3\hat{k}).$

Answer 1

$\overrightarrow{r}=\hat{i}-\hat{j}+\lambda (\hat{i}+\hat{j}+\hat{k})+\mu (\hat{i}-2\hat{j}+3\hat{k})$

Normal of plane$=\overline{a}\times \overline{b}(\because \overline{a}=\hat{i}+\hat{j}+\hat{k};\overline{b}=\hat{i}-2\hat{j}+3\hat{k})$

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

$\overline{n}=5\hat{i}-2\hat{j}-3\hat{k}$

$\therefore$ Let $\overline{r}=x\hat{i}-y\hat{j}-z\hat{k}$

$(\overline{r}-(\hat{i}-\hat{j}))\overline{n}=0$

$(x-1)5+(y+1)(-2)+z(-3)=0$

$5x-2y-3z=7$ is required plane equation in cartesian form.