Question

Find the equation of the plane passing through (a, b, c) and parallel to the plane $\overrightarrow{r}·\left(\hat{i}+\hat{j}+\hat{k}\right)=2.$

Answer 1

$$\begin{gathered} \text{Substituting}\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}\text{in the given equation of the plane, we get} \\ \left(x\hat{i}+y\hat{j}+z\hat{k}\right).\left(\hat{i}+\hat{j}+\hat{k}\right)=2 \\ \Rightarrow x+y+z-2=0...\left(1\right) \\ \text{The equation of a plane which is parallel to plane (1) is of the form} \\ x+y+z=k...\left(2\right) \\ \text{It is given that plane (2) is passing through the point}\left(a,b,c\right)\text{. So,} \\ a+b+c=k \\ \text{Substituting this value of}\mathit{\text{k}}\text{in (2), we get} \\ x+y+z=a+b+c,\text{which is the required equation of the plane.} \end{gathered}$$