Question

Find the vector equation of each one of following planes.
(i) 2x − y + 2z = 8
(ii) x + y − z = 5
(iii) x + y = 3

Answer 1

$$\begin{gathered} \left(i\right)\text{The given equation of plane is} \\ 2x-y+2z=8 \\ \Rightarrow \left(x\hat{i}+y\hat{j}+z\hat{k}\right).\left(2\hat{i}-\hat{j}+2\hat{k}\right)=8 \\ \Rightarrow \overrightarrow{r}.\left(2\hat{i}-\hat{j}+2\hat{k}\right)=8,\text{which is the vector equation of the plane.} \\ \text{(Because the vector equation of the plane is}\overrightarrow{r}\text{.}\overrightarrow{n}\text{=}\overrightarrow{a}\text{.}\overrightarrow{n}\text{)} \end{gathered}$$

$$\begin{gathered} \left(ii\right)\text{The}\text{given equation of plane is} \\ x+y-z=8 \\ \Rightarrow \left(x\hat{i}+y\hat{j}+z\hat{k}\right).\left(\hat{i}+\hat{j}-\hat{k}\right)=8 \\ \Rightarrow \overrightarrow{r}.\left(\hat{i}+\hat{j}-\hat{k}\right)=8,\text{which is the vector equation of the plane.} \\ \text{(Because the vector equation of the plane is}\overrightarrow{r}\text{.}\overrightarrow{n}\text{=}\overrightarrow{a}\text{.}\overrightarrow{n}\text{)} \end{gathered}$$

$$\begin{gathered} \left(iii\right)\text{The}\text{given equation of plane is} \\ x+y=3 \\ \Rightarrow \left(x\hat{i}+y\hat{j}+z\hat{k}\right).\left(\hat{i}+\hat{j}+0\hat{k}\right)=3 \\ \Rightarrow \left(x\hat{i}+y\hat{j}+z\hat{k}\right).\left(\hat{i}+\hat{j}\right)=3 \\ \Rightarrow \overrightarrow{r}.\left(\hat{i}+\hat{j}\right)=3,\text{which is the vector equation of the plane.} \\ \text{(Because the vector equation of the plane is}\overrightarrow{r}\text{.}\overrightarrow{n}\text{=}\overrightarrow{a}\text{.}\overrightarrow{n}\text{)} \end{gathered}$$