Question

Find the vector equation of the following planes in non-parametric form. \[\vec{r} = \left( \lambda - 2\mu \right) \hat{i} + \left( 3 - \mu \right) \hat{j}  + \left( 2\lambda + \mu \right) \hat{k} \]

Answer 1

` \text{ The given equation of the plane is } `

\[ \vec{r} = \left( \lambda - 2\mu \right) \hat{i}  + \left( 3 - \mu \right) \hat{j}  + \left( 2\lambda + \mu \right) \hat{k}  \]

\[ \Rightarrow \vec{r} = \left( 0 \hat{i} + 3 \hat{j}  + 0 \hat{k}  \right) + \lambda \left( \hat{i}  + 0 \hat{j}  + 2 \hat{k}  \right) + \mu \left( - 2 \hat{i}  - \hat{j}  + \hat{k}  \right)\]

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

\[\text{ Here } , \vec{a} = 0 \hat{i}  + 3 \hat{j}  + 0 \hat{k} ; \vec{b} = \hat{i}  + 0 \hat{j} + 2 \hat{k}  ; \vec{c} = - 2 \hat{i} - \hat{j}  + \hat{k} \]

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

\[ = \begin{vmatrix}\hat{i} & \hat{j}  & \hat{k}  \\ 1 & 0 & 2 \\ - 2 & - 1 & 1\end{vmatrix}\]

\[ = 2 \hat{i}  - 5 \hat{j} - \hat{k}  \]

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

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

\[ \Rightarrow \vec{r} . \left( 2 \hat{i}  - 5 \hat{j}  - \hat{k}  \right) = \left( 0 \hat{i}  + 3 \hat{j}  + 0 \hat{k}  \right) . \left( 2 \hat{i}  - 5 \hat{j}  - \hat{k}  \right)\]

\[ \Rightarrow \vec{r} . \left( 2 \hat{i}  - 5 \hat{j}  - \hat{k} \right) = 0 - 15 + 0\]

\[ \Rightarrow \vec{r} . \left( 2 \hat{i}  - 5 \hat{j}  - \hat{k} \right) + 15 = 0\]