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Question
Find the vector equations of the following planes in scalar product form \[\left( \vec{r} \cdot \vec{n} = d \right):\] \[\vec{r} = \left( 2 \hat{i} - \hat{k} \right) + \lambda \hat{i} + \mu\left( \hat{i} - 2 \hat{j} - \hat{k}
\right)\]
Solution
` \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} .`
\[\text{ Here } , \vec{a} = 2 \hat{i} + 0 \hat{j} - \hat{k} ; \vec{b} = \hat{i} ; \vec{c} = \hat{i} - 2 \hat{j} - \hat{k} \]
\[\text{ Normal vector } , \vec{n} = \vec{b} \times \vec{c} \]
\[ = \begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \\ 1 & 0 & 0 \\ 1 & - 2 & - 1\end{vmatrix}\]
\[ = 0 \hat{i} + \hat{j} - 2 \hat{k} \]
\[ = \hat{j} - 2 \hat{k} \]
\[ \text{ The vector equation of the plane in scalar product form is } \]
\[ \vec{r} . \vec{n} = \vec{a} . \vec{n} \]
\[ \Rightarrow \vec{r} . \left( \hat{j} - 2 \hat{k} \right) = \left( 2 \hat{i} + 0 \hat{j} - \hat{k} \right) . \left( \hat{j} - 2 \hat{k} \right)\]
\[ \Rightarrow \vec{r} . \left( \hat{j} - 2 \hat{k} \right) = 2\]
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