Question

Line $\overrightarrow{r}=(\hat{i}-\hat{j}+\hat{k})+t(2\hat{i}-\hat{j}+\hat{k})$ contained in a plane to which vector $\overrightarrow{n}=3\hat{i}-2\hat{j}+\lambda \hat{k}$ is normal. Find the value of $\lambda$. Also find the vector equation of the plane.

Answer 1

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

if this line is contained in a plane to which $\overrightarrow{n}=3\hat{i}-2\hat{j}+\lambda \hat{k}$ is normal.

then $(2\hat{i}-\hat{j}+\hat{k}).(3\hat{i}-2\hat{j}+\lambda \hat{k})=0\Rightarrow 6+2+\lambda =0\Rightarrow \lambda =-8\therefore \overrightarrow{n}=3\hat{i}-2\hat{j}-8\hat{k}$

$\because$ line is contained by the plane

$\therefore (\hat{i}-\hat{j}+\hat{k}$ lies in it .

$\therefore$ Equation of plane is

$3(x-1)-2(y-(-1\left)\right)-8(z-1)=0\Rightarrow 3(x-1)-2(y+1)-8(z-1)=0\Rightarrow 3x-3-2y-2-8z+8=0\Rightarrow 3x-2y-8z+3=0$