Question

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

Answer 1

$$\begin{gathered} \text{The equation of the given plane is} \\ \overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}+\mu \overrightarrow{c} \\ \text{So, the plane passes through the vector}\overrightarrow{a}\text{and parallel to the vectors}\overrightarrow{b}\text{and}\overrightarrow{c}\text{.} \\ \text{So, the plane passes through the vector}\overrightarrow{a}\text{whose normal vector is}\overrightarrow{b}\text{×}\overrightarrow{a}\text{(It means that}\overrightarrow{n}\text{=}\overrightarrow{b}\text{×}\overrightarrow{a}\text{)} \\ \text{So, the equation of the plane in scalar product form is} \\ \left(\overrightarrow{r}-\overrightarrow{a}\right).\overrightarrow{n}=0 \\ \Rightarrow \left(\overrightarrow{r}-\overrightarrow{a}\right).\left(\overrightarrow{b}\times \overrightarrow{c}\right)=0 \end{gathered}$$