Question

Write the vector equation of the line passing through the point (1, −2, −3) and normal to the plane $$\overrightarrow{r}\cdot (2\hat{i}+\hat{j}+2\hat{k})=5.$$

 

Answer 1

$$\text{ The required line is normal to the plane }\overrightarrow{r}.(2\hat{i}+\hat{j}+2\hat{k})=5\text{ and it is parallel to the normal vector of the plane. }$$

$$\text{ So, the required line is parallel to the vector }\overrightarrow{b}=2\hat{i}+\hat{j}+2\hat{k}$$

$$\text{ It is given that the line passes through the point }(1,-2,-3)\text{ whose position vector is given by }\overrightarrow{a}=\hat{i}-2\hat{j}-3\hat{k}.$$

$$\text{ We know that the equation of the line passing through the point whose position vector is }\overrightarrow{a}\text{ and parallel to the vector }\overrightarrow{b}\text{ is given by }$$

$$\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}$$

$$\Rightarrow \overrightarrow{r}=(\hat{i}-2\hat{j}-3\hat{k})+\lambda (2\hat{i}+\hat{j}+2\hat{k})$$