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Question
Find the vector equation of the following planes in non-parametric form. \[\vec{r} = \left( \lambda - 2\mu \right) \hat{i} + \left( 3 - \mu \right) \hat{j} + \left( 2\lambda + \mu \right) \hat{k} \]
Solution
` \text{ The given equation of the plane is } `
\[ \vec{r} = \left( \lambda - 2\mu \right) \hat{i} + \left( 3 - \mu \right) \hat{j} + \left( 2\lambda + \mu \right) \hat{k} \]
\[ \Rightarrow \vec{r} = \left( 0 \hat{i} + 3 \hat{j} + 0 \hat{k} \right) + \lambda \left( \hat{i} + 0 \hat{j} + 2 \hat{k} \right) + \mu \left( - 2 \hat{i} - \hat{j} + \hat{k} \right)\]
\[ \text{ We know that the equation } \vec{r} = \vec{a} + \lambda \vec{b} + \mu \vec{c} \text{ represents a plane passing through a point whose position vector is }\vec{a} \text{ and parallel to the vectors } \vec{b} \text{ and } \vec{c} .\]
\[\text{ Here } , \vec{a} = 0 \hat{i} + 3 \hat{j} + 0 \hat{k} ; \vec{b} = \hat{i} + 0 \hat{j} + 2 \hat{k} ; \vec{c} = - 2 \hat{i} - \hat{j} + \hat{k} \]
\[ \text{ Normal vector } , \vec{n} = \vec{b} \times \vec{c} \]
\[ = \begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \\ 1 & 0 & 2 \\ - 2 & - 1 & 1\end{vmatrix}\]
\[ = 2 \hat{i} - 5 \hat{j} - \hat{k} \]
\[ \text{ The vector equation of the plane in scalar product form is } \]
\[ \vec{r} . \vec{n} = \vec{a} . \vec{n} \]
\[ \Rightarrow \vec{r} . \left( 2 \hat{i} - 5 \hat{j} - \hat{k} \right) = \left( 0 \hat{i} + 3 \hat{j} + 0 \hat{k} \right) . \left( 2 \hat{i} - 5 \hat{j} - \hat{k} \right)\]
\[ \Rightarrow \vec{r} . \left( 2 \hat{i} - 5 \hat{j} - \hat{k} \right) = 0 - 15 + 0\]
\[ \Rightarrow \vec{r} . \left( 2 \hat{i} - 5 \hat{j} - \hat{k} \right) + 15 = 0\]
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