Question

The plane $\overrightarrow{r}\cdot \overrightarrow{n}=q$ will contain the line $\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}$, if-

• A. $\overrightarrow{b}\cdot \overrightarrow{n}\ne 0,\overrightarrow{a}\cdot \overrightarrow{n}\ne q$
• B. $\overrightarrow{b}\cdot \overrightarrow{n}=0,\overrightarrow{a}\cdot \overrightarrow{n}\ne q$
• C. $\overrightarrow{b}\cdot \overrightarrow{n}=0,\overrightarrow{a}\cdot \overrightarrow{n}=q$
• D. $\overrightarrow{b}\cdot \overrightarrow{n}\ne 0,\overrightarrow{a}\cdot \overrightarrow{n}=q$

Answer 1

The correct option is C $\overrightarrow{b}\cdot \overrightarrow{n}=0,\overrightarrow{a}\cdot \overrightarrow{n}=q$
Normal of the plane $\overrightarrow{r}\cdot \overrightarrow{n}=q$ is $\overrightarrow{n}$ and direction ratio of the line $\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}$ is $\overrightarrow{b}$
Since, line lies in the plane, normal and direction ratios should be perpendicular.
Therefore, $\overrightarrow{b}\cdot \overrightarrow{n}=0$
Also position vector $\overrightarrow{a}$ should lie on plane $\overrightarrow{r}\cdot \overrightarrow{n}=q$
Therefore, $\overrightarrow{a}\cdot \overrightarrow{n}=q$
Ans: C