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Question
Write the equation of the plane \[\vec{r} = \vec{a} + \lambda \vec{b} + \mu \vec{c}\] in scalar product form.
Solution
\[\text{ The equation of the given plane is } \]
\[ \vec{r} = \vec{a} + \lambda \vec{b} + \mu \vec{c} \]
\[\text{ So, the plane passes through the vector } \vec{a} \text{ and parallel to the vectors } \vec{b} \text{ and } \vec{c} .\]
\[\text{ So, the plane passes through the vector }\vec{a} \text{ whose normal vector is } \vec{b} \times \vec{a} (\text{ It means that } \vec{n} = \vec{b} \times \vec{a} )\]
\[\text{ So, the equation of the plane in scalar product form is } \]
\[\left( \vec{r} - \vec{a} \right) . \vec{n} = 0\]
\[ \Rightarrow \left( \vec{r} - \vec{a} \right) . \left( \vec{b} \times \vec{c} \right) = 0\]
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