R D Sharma Solutions for Chapter: The Plane, Exercise 13: EXERCISE

Show that the lines $\overrightarrow{r}=\left(2\hat{j}-3\hat{k}\right)+\lambda \left(\hat{i}+2\hat{j}+3\hat{k}\right)$ and $\overrightarrow{r}=\left(2\hat{i}+6\hat{j}+3\hat{k}\right)+\mu \left(2\hat{i}+3\hat{j}+4\hat{k}\right)$ are coplanar. Also, find the equation of the plane containing them.

The equation of the plane which contains two parallel lines $\frac{x-4}{1}=\frac{y-3}{-4}=\frac{z-2}{5}$ and $\frac{x-3}{1}=\frac{y+2}{-4}=\frac{z}{5}$ is $mx-y-3z=35$, find $m$?

Find the vector equation of the plane passing through three points with position vectors $\hat{i}+\hat{j}-2\hat{k}, 2\hat{i}-\hat{j}+\hat{k}$ and $\hat{i}+2\hat{j}+\hat{k}.$ Also, find the coordinates of the point of intersection of this plane and the line $\overrightarrow{r}=3\hat{i}-\hat{j}-\hat{k}+\lambda \left(2\hat{i}-2\hat{j}+\hat{k}\right).$

The equation of a plane which passes through the point$(3, 2, 0)$ and contains the line $\frac{x-3}{1}=\frac{y-6}{5}=\frac{z-4}{4}$ is $x+my+z=1$, find $m$?

Show that the lines $\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-5}{5}$ and $\frac{x+1}{-1}=\frac{y-2}{2}=\frac{z-5}{5}$ are coplanar. Hence, find the equation of the plane containing these lines.

Find the values of $\lambda$ for which the lines $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z+3}{{\lambda }^2}$ and $\frac{x-3}{1}=\frac{y-2}{{\lambda }^2}=\frac{z-1}{2}$ are coplanar.

If the lines $x=5, \frac{y}{3-\alpha }=\frac{z}{-2}$ and $x=\alpha , \frac{y}{-1}=\frac{z}{2-\alpha }$ are coplanar, find the values of $\alpha$.

If the straight lines$\frac{x-1}{2}=\frac{y+1}{k}=\frac{z}{2}$ and $\frac{x+1}{2}=\frac{y+1}{2}=\frac{z}{k}$ are coplanar, find the equations of the planes containing them.