Question

Find the equation of the plane in scalar product form $\hat{r}=\left(2\hat{i}+\hat{k}\right)+\lambda \hat{i}+\mu \left(\hat{i}+2\hat{j}-3\hat{k}\right)$

Answer 1

Given plane equation is ,

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

which is passes through point with position vector:

$\overline{OA}=2\hat{i}+\hat{k}$

and parallel to vector :

$\overline{b}=\hat{i}$

$\overline{c}=\hat{i}+2\hat{j}-3\hat{k}$

the quation of plane in scaler triple product fron is

$\begin{array}{c}\left[\overline{r}-\overline{OA}\overline{b}\overline{c}\right]=0\\ \left(\begin{array}{ccc}x-1& y-2& z+1\\ 1& 0& 0\\ 1& 2& -3\end{array}\right)=0\\ (x-2)(0-0)-\left(y\right)(-3)+(z-1)\left(2\right)=0\\ \therefore 3y+2z-2=o\end{array}$