Question

Write the equation of the plane  \[\vec{r} = \vec{a} + \lambda \vec{b} + \mu \vec{c}\]   in scalar product form.

 

Answer 1

\[\text{ The equation of the given plane is } \]

\[ \vec{r} = \vec{a} + \lambda \vec{b} + \mu \vec{c} \]

\[\text{ So, the plane passes through the vector } \vec{a} \text{ and parallel to the vectors } \vec{b} \text{ and } \vec{c} .\]

\[\text{ So, the plane passes through the vector }\vec{a} \text{ whose normal vector is } \vec{b} \times \vec{a} (\text{ It means that }  \vec{n} = \vec{b} \times \vec{a} )\]

\[\text{ So, the equation of the plane in scalar product form is } \]

\[\left( \vec{r} - \vec{a} \right) . \vec{n} = 0\]

\[ \Rightarrow \left( \vec{r} - \vec{a} \right) . \left( \vec{b} \times \vec{c} \right) = 0\]