Embibe Experts Solutions for Exercise 6: Assignment

$\left(\mathrm{i}\right)$ Find the point where the plane $\overrightarrow{r}·\left(\hat{i}+2\hat{j}+\hat{k}\right)=4$ meets the line $\overrightarrow{r}=\left(2\hat{i}+\hat{j}-\hat{k}\right)+t\left(\hat{i}+\hat{j}+2\hat{k}\right)$.

$\left(\mathrm{ii}\right)$ Find the angle that the line makes with the plane.

Find the equations of straight line perpendicular to the lines $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$ and $\frac{x}{1}=\frac{y-7}{-3}=\frac{z+7}{2}$, and passing through their point of intersection.

Prove that the line passing through $\left(1,-2,0\right)$ having direction ratios $2,3,6$ and the line $3\left(x+3\right)=2\left(y-5\right)=z+1$ are parallel. Find the distance between the lines.

Find the shortest distance between the lines $\frac{x-3}{2}=\frac{y-4}{1}=\frac{z+1}{-3}$ and $\frac{x-1}{-1}=\frac{y-3}{3}=\frac{z-1}{2}$ also find the equation of line of shortest distance.

Find the equation of line through the intersection of two planes $x+y-2z=1$, and $x+3y-z=4$, and also find its perpendicular distance from $\left(1,0,1\right)$.

Find the linear equation describing the plane perpendicular to the line of intersection of the planes $x+y-2z=4$ and $3x-2y+z=1$ and which passes through $\left(6,0,2\right)$, also find foot of perpendicular from origin to this plane.

The points $A\left(1,1,5\right), B\left(2,2,1\right), C\left(1,-2,2\right)$ and $D\left(-2,1,2\right)$ are vertices of tetrahedron. Find the equation of the line through $A$ perpendicular to face $BCD$ and distance of $A$ from this face. Also find the shortest distance between skew lines $AD$ and $BC$.

By computing the shortest distance, determine whether the following pairs of line intersect or not :

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