Question

The equation of the plane passing through (1, 2, 3) and parallel to the plane $2x+3y-4z=0$ is __________.

Answer 1

For plane normal is given by (2, 3, −4) [∴being parallel to the plane 2x + 3y − 4z = 0]

$$\begin{gathered} \mathrm{i}.\mathrm{e}.\overrightarrow{n}=\left(2,3,-4\right)\mathrm{and}\overrightarrow{\alpha }=\left(1,2,3\right)\left[\because \mathrm{given}\mathrm{that}\mathrm{plane}\mathrm{passes}\mathrm{through}\left(1,2,3\right)\right] \\ \therefore \mathrm{For}\overrightarrow{r}=x\hat{i}+y\hat{j}+2\hat{k}, \\ \mathrm{Equation}\mathrm{of}\mathrm{plane}\mathrm{is}\left(\overrightarrow{r}-\overrightarrow{\alpha }\right).\overrightarrow{n}=0 \\ \mathrm{i}.\mathrm{e}.\overrightarrow{r}.\overrightarrow{n}=\overrightarrow{\alpha }.\overrightarrow{n} \\ \mathrm{i}.\mathrm{e}.\left(x\hat{i}+y\hat{j}+z\hat{k}\right),\left(2\hat{i}+3\hat{j}-4\hat{k}\right)=\left(\hat{i}+2\hat{j}+3\hat{k}\right).\left(2\hat{i}+3\hat{j}-4\hat{k}\right) \\ \mathrm{i}.\mathrm{e}.2x+3y-4z=2+6-12 \\ \mathrm{i}.\mathrm{e}.2x+3y-4z=-4 \end{gathered}$$
i.e. 2x + 3y – 4z + 4 = 0 is the equation of plane which passes through (1, 2, 3) and is parallel to 2x + 3y – 4z = 0