Question

Find the line which is parallel to both the planes
$\overrightarrow{r}\cdot {\overrightarrow{n}}_1=q_1,\overrightarrow{r}\cdot {\overrightarrow{n}}_2=q_2$ and passing through point $a$.

• A. $r=a+\lambda ({\overrightarrow{n}}_1\times {\overrightarrow{n}}_2)$
• B. $r=a\cdot \lambda ({\overrightarrow{n}}_1\times {\overrightarrow{n}}_2)$
• C. $r=a+n_1\times n_2$
• D. $r\cdot a=a+n_1+n_2$

Answer 1

Answer: (A)

$r=a+\lambda ({\overrightarrow{n}}_1\times {\overrightarrow{n}}_2)$

The required line is parallel to both the planes.

Hence its direction vector is perpendicular to the normals of both the planes.

Thus the direction vector for the line will be

$\overrightarrow{n}=\overrightarrow{n_1}\times \overrightarrow{n_2}$

Now the line passes through $\overrightarrow{a}$.

Hence the equation of the line will be given by

$\overrightarrow{r}=a+\lambda \left(\overrightarrow{n}\right)$ where $\lambda$ is a scalar constant.

By replacing the value of $\overrightarrow{n}$ we get

$\overrightarrow{r}=a+\lambda (\overrightarrow{n_1}\times \overrightarrow{n_2})$.