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

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