Question

Find the equation of the plane passing through the point (2, 3, 1), given that the direction ratios of the normal to the plane are proportional to 5, 3, 2.

Answer 1

$$\begin{gathered} \text{We know that the vector equation of the plane passing through a point}\overrightarrow{a}\text{and normal to}\overrightarrow{n}\text{is} \\ \overrightarrow{r}.\overrightarrow{n}=\overrightarrow{a}.\overrightarrow{n} \\ \text{Substituting}\overrightarrow{a}\text{= 2}\hat{i}\text{+ 3}\hat{j}\text{+}\hat{k}\text{and}\overrightarrow{n}\text{= 5}\hat{i}\text{+ 3}\hat{j}\text{+ 2}\hat{k}\text{, we get} \\ \overrightarrow{r}.\left(5\hat{i}\text{+ 3}\hat{j}\text{+ 2}\hat{k}\right)=\left(2\hat{i}\text{+ 3}\hat{j}\text{+}\hat{k}\right).\left(5\hat{i}\text{+ 3}\hat{j}\text{+ 2}\hat{k}\right) \\ \overrightarrow{r}.\left(5\hat{i}\text{+ 3}\hat{j}\text{+ 2}\hat{k}\right)=10+9+2 \\ \overrightarrow{r}.\left(5\hat{i}\text{+ 3}\hat{j}\text{+ 2}\hat{k}\right)=21 \\ \text{Substituting}\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}\text{in the vector equation, we get} \\ \left(x\hat{i}+y\hat{j}+z\hat{k}\right).\left(5\hat{i}\text{+ 3}\hat{j}\text{+ 2}\hat{k}\right)=21 \\ \Rightarrow 5x+3y+2z=21 \end{gathered}$$