Question

Write the position vector of the point where the line $\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}$ meets the plane $\overrightarrow{r}.\overrightarrow{n}=0.$

Answer 1

$$\begin{gathered} \text{Given equation of the line is} \\ \overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}...\left(1\right) \\ \text{Given equation of the plane is} \\ \overrightarrow{r}.\overrightarrow{n}=0 \\ \Rightarrow \left(\overrightarrow{a}+\lambda \overrightarrow{b}\right).\overrightarrow{n}=0[\text{From (1)]} \\ \Rightarrow \overrightarrow{a}.\overrightarrow{n}+\lambda \overrightarrow{b}.\overrightarrow{n}=0 \\ \Rightarrow \lambda =-\left(\frac{\overrightarrow{a}.\overrightarrow{n}}{\overrightarrow{b}.\overrightarrow{n}}\right) \\ \text{Substituting this in (1), we get} \\ \overrightarrow{r}=\overrightarrow{a}-\left(\frac{\overrightarrow{a}.\overrightarrow{n}}{\overrightarrow{b}.\overrightarrow{n}}\right)\overrightarrow{b},\text{which is the required position vector that lies both on the line and the plane.} \end{gathered}$$