Equation of a Line in Space

Find the length and the equation of the common perpendicular to the lines: $\frac{x+3}{-4}=\frac{y-6}{3}=\frac{z}{2}\text{ and }\frac{x+2}{-4}=\frac{y}{1}=\frac{z-7}{1}$.

Point of intersection of the lines $\frac{x-3}{3}=\frac{y-3}{-1},z-1=0 \mathrm{and} \frac{{\displaystyle x-6}}{{\displaystyle 2}}=\frac{{\displaystyle z-1}}{{\displaystyle 3}},y-2=0$ is

Point of intersection of the lines $\overrightarrow{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda (3\hat{i}-\hat{j})$ and $\overrightarrow{r}=(4\hat{i}-\hat{k})+\mu (2\hat{i}+3\hat{k})$ is

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

The equation of the line in vector form passing through the point $\left(4,-2,5\right)$ and parallel to the vector $3\hat{i}-\hat{j}+2\hat{k}$  is $\overrightarrow{r}=\left(3\hat{i}-\hat{j}+2\hat{k}\right)+\lambda \left(4\hat{i}-2\hat{j}+5\hat{k}\right)$

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 

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

If two lines $\overrightarrow{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda (3\hat{i}-\hat{j})$ and $\overrightarrow{r}=(4\hat{i}-\hat{k})+\mu (2\hat{i}+3\hat{k})$ are given then

Find the vector equation of the line passing through the points $(3,4,-7)$ and $(6,-1,1)$

Equation of the line passing through the point $(3,1,2)$ and perpendicular to the lines $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}$ and $\frac{x}{3}=\frac{y}{2}=\frac{z}{5}$ is

If the point $A(\lambda ,5,-2)$ lies on the line $\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}$, then the value of $\lambda$ is

The Cartesian equation of the line passing through the points $A(4,2,1)$ and $B(2,-1,3)$ is 

Find the vector equation of a line passing through a point with position vector $2\hat{i}-\hat{j}+\hat{k}$  and parallel to the line joining the points $-\hat{i}+4\hat{j}+\hat{k}$ and $\hat{i}+2\hat{j}+2\hat{k}$.

Find the equation of line equally inclined to coordinate axes and passes through $\left(-5,1,-2\right)$.