Question

Show that the points $\hat{i}-\hat{j}+3\hat{k}\mathrm{and}3\hat{i}+3\hat{j}+3\hat{k}$ are equidistant from the plane $\overrightarrow{r}·\left(5\hat{i}+2\hat{j}-7\hat{k}\right)+9=0.$

Answer 1

$$\begin{gathered} \text{The given plane is} \\ \overrightarrow{r}.\left(5\hat{i}+2\hat{j}-7\hat{k}\right)+9=0 \\ \Rightarrow \overrightarrow{r}.\left(-5\hat{i}-2\hat{j}+7\hat{k}\right)=9 \\ \text{We know that the perpendicular distance of a point}\mathit{\text{P}}\text{of position vector}\overrightarrow{a}\text{from the plane}\overrightarrow{r}\text{.}\overrightarrow{n}\text{=}\mathit{\text{d}}\text{is given by} \\ p=\frac{\left|\overrightarrow{a}.\overrightarrow{n}-d\right|}{\left|\overrightarrow{n}\right|} \\ \mathbf{\text{Finding the distance from}}\hat{\mathbf{i}}\mathbf{-}\hat{\mathbf{j}}\mathbf{+}\mathbf{3}\mathbf{}\hat{\mathbf{k}}\mathbf{}\mathbf{\text{to the given plane}} \\ \text{Here,}\overrightarrow{a}=\hat{i}-\hat{j}+3\hat{k};\overrightarrow{n}=-5\hat{i}-2\hat{j}+7\hat{k};d=9 \\ \text{So, the required distance}\mathit{\text{p}}\text{is given by} \\ p=\frac{\left|\left(\hat{i}-\hat{j}+3\hat{k}\right).\left(-5\hat{i}-2\hat{j}+7\hat{k}\right)-9\right|}{\left|-5\hat{i}-2\hat{j}+7\hat{k}\right|} \\ =\frac{\left|-5+2+21-9\right|}{\sqrt{25+4+49}} \\ =\frac{\left|9\right|}{\sqrt{78}} \\ =\frac{9}{\sqrt{78}}\text{units ... (1)} \\ \mathbf{\text{Finding the distance from 3}}\hat{i}\mathbf{\text{+}}\mathbf{3}\mathit{}\hat{j}\mathbf{+}\mathbf{3}\mathit{}\hat{k}\mathbf{}\mathbf{\text{to the given plane}} \\ \text{Here,}\overrightarrow{a}=3\hat{i}+3\hat{j}+3\hat{k};\overrightarrow{n}=-5\hat{i}-2\hat{j}+7\hat{k};d=9 \\ \text{So, the required distance}\mathit{\text{p}}\text{is given by} \\ p=\frac{\left|\left(3\hat{i}+3\hat{j}+3\hat{k}\right).\left(-5\hat{i}-2\hat{j}+7\hat{k}\right)-9\right|}{\left|3\hat{i}+3\hat{j}+3\hat{k}\right|} \\ =\frac{\left|-15-6+21-9\right|}{\sqrt{25+4+49}} \\ =\frac{\left|-9\right|}{\sqrt{78}} \\ =\frac{9}{\sqrt{78}}\text{units ... (2)} \\ \text{From (1) and (2), we can say that the given points are equidistant from the given plane.} \end{gathered}$$