Lines in 3D

The distance of the point $A\left(-2,3,1\right)$ from the line $PQ$ through $P\left(-3,5,2\right)$ which make equal angle with axes is 

Find the equation of a plane passing through the points $A\left(2,1,2\right)$ and $B\left(4,-2,1\right)$ and perpendicular to plane $\overrightarrow{r}·\left(\hat{i}-2\hat{k}\right)=5$. Also, find the coordinates of the point, where the line passing through the points $\left(3,4,1\right)$ and $\left(5,1,6\right)$ crosses the plane thus obtained.

The length of the perpendicular from the point $\left(1,2,3\right)$ to the line $\frac{-x+6}{-3}=\frac{y-7}{2}=\frac{z-7}{-2}$ is _________.

The distance of the point $(1,2,3)$ from the plane $x+y+z=2$ measured parallel to the line $\frac{x+1}{-1}=\frac{y}{-2}=\frac{z-3}{1}$ is

If the image of the line $L_1: \frac{x-1}{1}=\frac{y-1}{2}=\frac{z-1}{3}$ with respect to the line $L_2: \frac{x-1}{2}=\frac{y-1}{3}=\frac{z-1}{1}$ is the line $L_3$, then equation of the line $L_3$ is

The perpendicular distance of the point $P(2, 1, 3)$ from the line $\frac{x-1}{3}=\frac{y-3}{1}=\frac{z-4}{-5}$ is

The coordinate of a point on the line $\frac{x-2}{1}=\frac{y+1}{-1}=\frac{z}{3}$ at a distance $\sqrt{21}$ from the point $\left(-1, -1, 1\right)$ is -

The length of the perpendicular from $A(2,-3,1)$ on the line$\frac{x+1}{2}=\frac{y-3}{3}=\frac{z+2}{-1}$ is

The vector equation of line passing through the point $\left(5,4,3\right)$ and having direction ratios $-3,4,2$ is

Length of the perpendicular from the point $\left(5,4,-1\right)$ to the line $\overrightarrow{r}=\hat{i}+\lambda (2\hat{i}+9\hat{j}+5\hat{k})$ is

If the points $A(-1,3,2), B(-4,2,-2)$ and $\mathrm{C}(5,5,\lambda )$ are collinear, find the value of $\lambda$.

Find the vector equation of a line passing through the point $\hat{i}+2\hat{j}+3\hat{k}$, and perpendicular to the vectors $\hat{i}+\hat{j}+\hat{k}$ and $2\hat{i}-\hat{j}+\hat{k}$.

Equation of line equally inclined to coordinate axes and passes through $\left(-5,1,-2\right)$ is

If the points $A\left(5,5,\lambda \right),B\left(-1,3,2\right)$ and $C\left(-4,2,-2\right)$ are collinear, then the value of $\lambda$ is

The equation of the line in Cartesian form passing through the point with position vector $2\hat{i}-\hat{j}-4\hat{k}$  and is in the direction of $\hat{i}-2\hat{j}+\hat{k}$ is 

The image of the point (1, 6, 3) in the line $\overrightarrow{r}=\left(\widehat{j}+2\widehat{k}\right)+\lambda \left(\widehat{i}+2\widehat{j}+3\widehat{k}\right)$ is