Question

Find the vector equation of a plane passing through a point with position vector \[2 \hat{i} - \hat{j} + \hat{k} \] and perpendicular to the vector  \[4 \hat{i} + 2 \hat{j} - 3 \hat{k} .\] 

Answer 1

\[\text{ We know that the vector equation of the plane passing through a point }  \vec{a} \text{ and normal to } \vec{n} is\]
\[ \vec{r} . \vec{n} = \vec{a} . \vec{n} \]
\[ \text{ Substituting } \vec{a} = 2 \hat{ i} - \hat{j} + \hat{k}  \text{ and }  \vec{n} = 4 \hat{i}  + 2 \hat{j}  - 3 \hat{k}  , \text{ we get } \]
\[ \vec{r} . \left( 4 \hat{i}  + 2 \hat{j}  - 3 \hat{k}  \right) = \left( 2 \hat{i}  - \hat{j}  + \hat{k}  \right) . \left( 4 \hat{i}  + 2 \hat{j}  - 3 \hat{k}  \right)\]
\[ \Rightarrow \vec{r} . \left( 4 \hat{i} + 2 \hat{j}  - 3 \hat{k}  \right) = 8 - 2 - 3\]
\[ \Rightarrow \vec{r} . \left( 4 \hat{i}  + 2 \hat{j}  - 3 \hat{k}  \right) = 3\]