Question

Find the vector equation of the plane passing through the points P (2, 5, −3), Q (−2, −3, 5) and R (5, 3, −3).

Answer 1

$$\begin{gathered} \text{The required plane passes through the point}\mathit{\text{P}}\text{(2, 5, -3) whose position vector is}\overrightarrow{a}\text{=2}\hat{i}\text{+5}\hat{j}\text{-3}\hat{k}\text{and is normal to the vector}\overrightarrow{n}\text{given by} \\ \overrightarrow{n}=\overrightarrow{PQ}\times \overrightarrow{PR.} \\ \text{Clearly,}\overrightarrow{PQ}=\overrightarrow{OQ}-\overrightarrow{OP}=\left(-2\hat{i}-3\hat{j}+5\hat{k}\right)-\left(2\hat{i}\text{+ 5}\hat{j}\text{- 3}\hat{k}\right)=-4\hat{i}-8\hat{j}+8\hat{k} \\ \overrightarrow{PR}=\overrightarrow{OR}-\overrightarrow{OP}=\left(5\hat{i}+3\hat{j}-3\hat{k}\right)-\left(2\hat{i}\text{+ 5}\hat{j}\text{- 3}\hat{k}\right)=3\hat{i}-2\hat{j}-0\hat{k} \\ \overrightarrow{n}=\overrightarrow{PQ}\times \overrightarrow{PR}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ -4& -8& 8\\ 3& -2& 0\end{array}\right|=16\hat{i}+24\hat{j}+32\hat{k} \\ \text{The vector equation of the required plane is} \\ \overrightarrow{r}.\overrightarrow{n}=\overrightarrow{a}.\overrightarrow{n} \\ \Rightarrow \overrightarrow{r}.\left(16\hat{i}+24\hat{j}+32\hat{k}\right)=\left(2\hat{i}\text{+5}\hat{j}\text{-3}\hat{k}\right).\left(16\hat{i}+24\hat{j}+32\hat{k}\right) \\ \Rightarrow \overrightarrow{r}.\left[8\left(2\hat{i}+3\hat{j}+4\hat{k}\right)\right]=32+120-96 \\ \Rightarrow \overrightarrow{r}.\left[8\left(2\hat{i}+3\hat{j}+4\hat{k}\right)\right]=56 \\ \Rightarrow \overrightarrow{r}.\left(2\hat{i}+3\hat{j}+4\hat{k}\right)=7 \end{gathered}$$