Maharashtra Board Solutions for Chapter: Line and Plane, Exercise 3: Miscellaneous Exercise 6A

Find the co-ordinates of the foot of the perpendicular drawn from the point $\left(0,2,3\right)$ to the line $\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}$.

By computing the shortest distance determine whether following lines intersect each other.

$\overrightarrow{r}=\left(\hat{i}+\hat{j}-\hat{k}\right)+\lambda \left(2\hat{i}-\hat{j}+\hat{k}\right)$ and $\overrightarrow{r}=\left(2\hat{i}+2\hat{j}-3\hat{k}\right)+\mu \left(\hat{i}+\hat{j}-2\hat{k}\right)$

By computing the shortest distance determine whether following lines intersect each other.

$\frac{x-5}{4}=\frac{y-7}{5}=\frac{z+3}{5}$ and $x-6=y-8=z+2$

If lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-2}{1}=\frac{y+m}{2}=\frac{z-2}{1}$ intersect each other then find $m$.

Find the direction cosines of the line $\overrightarrow{r}=\left(-2\hat{i}+\frac{5}{2}\hat{j}-\hat{k}\right)+\lambda \left(2\hat{i}+3\hat{j}\right)$.

Find the Cartesian equation of the line passing through the origin which is perpendicular to $x-1=y-2=z-1$ and intersects the $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$.

Write the vector equation of the line whose Cartesian equations are $y=2$ and $4x-3z+5=0$.

Find the co-ordinates of points on the line $\frac{x-1}{1}=\frac{y-2}{-2}=\frac{z-3}{2}$ which are at the distance $3$ unit from the base point $A\left(1,2,3\right)$.