Question

Find the vector equations of the following planes in scalar product form  \[\left( \vec{r} \cdot \vec{n} = d \right):\]\[\vec{r} = \left( \hat{i}  + \hat{j}  \right) + \lambda\left( \hat{i}  + 2 \hat{j}  - \hat{k}  \right) + \mu\left( - \hat{i}  + \hat{j} - 2 \hat{k} \right)\]

Answer 1

` \text{ We know that the equation }  \vec{r} = \vec{a} + \lambda \vec{b} + \mu \vec{c} \text{ represents a plane passing through a point whose position vector is } \vec{a} \text{ and parallel to the vectors }  \vec{b} \text{ and } \vec{c} .`

\[\text{ Here } , \vec{a} = \hat{i} + \hat{j} + 0 \hat{k} ; \vec{b} = \hat{i}  + 2 \hat{j}  - \hat{k}  ; \vec{c} = - \hat{i}  + \hat{j} - 2 \hat{k}  \]

\[\text{ Normal vector } , \vec{n} = \vec{b} \times \vec{c} \]

\[ = \begin{vmatrix}\hat{i}  & \hat{j} & \hat{k}  \\ 1 & 2 & - 1 \\ - 1 & 1 & - 2\end{vmatrix}\]

\[ = - 3 \hat{i}  + 3 \hat{j}  + 3 \hat{k} \]

\[\text{ The vector equation of the plane in scalar product form is } \]

\[ \vec{r} . \vec{n} = \vec{a} . \vec{n} \]

\[ \Rightarrow \vec{r} . \left( - 3 \hat{i}  + 3 \hat{j}  + 3 \hat{k} \right) = \left( \hat{i}  + \hat{j} + 0 \hat{k}  \right) . \left( - 3 \hat{i}  + 3 \hat{j}  + 3 \hat{k}  \right)\]

\[ \Rightarrow \vec{r} . \left( - 3 \hat{i}  + 3 \hat{j} + 3 \hat{k} \right) = - 3 + 3\]

\[ \Rightarrow \vec{r} . \left[ 3 \left( - \hat{i} + \hat{j}  + \hat{k}  \right) \right] = 0\]

\[ \Rightarrow \vec{r} . \left( - \hat{i}  + \hat{j}  + \hat{k} \right) = 0\]