Question

Find the vector equation of the plane, passing through the point (a, b, c) and parallel to the plane \[\vec{r} . \left( \hat{i}  + \hat{j}  + \hat{k}  \right) = 2\]

 

Answer 1

The required plane passes through \[a \hat{i} + b \hat{j} + c \hat{k} \] and is parallel to the plane  \[\vec{r} . \left( \hat{i} + \hat{j} + \hat{k} \right) = 2\] So, it is normal to the vector  \[\hat{i}  + \hat{j} + \hat{k} \] which is normal to the given plane.
Hence, the vector equation of the required plane is

\[\left[ \vec{r} - \left( a \hat{i} + b \hat{j} + c \hat{k} \right) \right] . \left( \hat{i} + \hat{j} + \hat{k}  \right) = 0 \left[ \left( \vec{r} - \vec{a} \right) . \vec{n} = 0 \right]\]
\[ \Rightarrow \vec{r} . \left( \hat{i}  + \hat{j}  + \hat{k} \right) = \left( a \hat{i} + b \hat{j} + c \hat{k}  \right) . \left( \hat{i}  + \hat{j}  + \hat{k}  \right)\]
\[ \Rightarrow \vec{r} . \left( \hat{i}  + \hat{j}  + \hat{k}  \right) = a + b + c\]

Thus, the vector equation of the required plane is

\[\vec{r} . \left( \hat{i} + \hat{j} + \hat{k}  \right) = a + b + c\]