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Question
Find the vector equation of the following planes in non-parametric form. \[\vec{r} = \left( 2 \hat{i} + 2 \hat{j} - \hat{k} \right) + \lambda\left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) + \mu\left( 5 \hat{i} - 2 \hat{j} + 7 \hat{k} \right)\]
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} .`
\[\text{ Here }, \vec{a} = 2 \hat{i} + 2 \hat{j} - \hat{k} ; \vec{b} = \hat{i} + 2 \hat{j} + 3 \hat{k} ; \vec{c} = 5 i - 2 \hat{j} + 7 \hat{k} \]
\[\text{ Normal vector } , \vec{n} = \vec{b} \times \vec{c} \]
\[ = \begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 3 \\ 5 & - 2 & 7\end{vmatrix}\]
\[ = 20 \hat{i} + 8 \hat{j} - 12 \hat{k} \]
\[ \text{ The vector equation of the plane in scalar product form is } \]
\[ \vec{r} . \vec{n} = \vec{a} . \vec{n} \]
\[ \Rightarrow \vec{r} . \left( 20 \hat{i} + 8 \hat{j} - 12 \hat{k} \right) = \left( 2 \hat{i} + 2 \hat{j} - \hat{k} \right) . \left( 20 \hat{i} + 8 \hat{j} - 12 \hat{k} \right)\]
\[ \Rightarrow \vec{r} . \left( 4 \left( 5 \hat{i} + 2 \hat{j} - 3 \hat{k} \right) \right) = 40 + 16 + 12\]
\[ \Rightarrow \vec{r} . \left( 4 \left( 5 \hat{i} + 2 \hat{j} - 3 \hat{k} \right) \right) = 68\]
\[ \Rightarrow \vec{r} . \left( 5 \hat{i} + 2 \hat{j} - 3 \hat{k} \right) = 17\]
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