Question

Write the equation of the plane containing the lines \[\vec{r} = \vec{a} + \lambda \vec{b} \text{ and }  \vec{r} = \vec{a} + \mu \vec{c} .\]

 

Answer 1

\[\text{ The given plane passes through the lines } \]

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

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

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

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

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

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