Shortest Distance between Two Lines

If the two lines $\frac{x-2m}{2m+5}=\frac{y}{8m}=\frac{z-4}{2}$ and $\frac{x-2}{m-2}=\frac{y}{-1}=\frac{z-2m}{1-3m}$ are parallel for some $m\in R$, then the distance between them is:

What is the condition to check whether two vectors are intersecting or not $?$

Two submarines are moving along the lines $\frac{x}{1}=\frac{y}{-1}=\frac{z}{1} \mathrm{and} \frac{x-1}{0}=\frac{y+1}{-2}=\frac{z}{1}$. At which points(s) on their way they are nearest each other?

In a realistic city model, a metro track and a road above it are straight lines represented by $\hat{i}-2\hat{j}+2\hat{k}+\lambda \left(\hat{i}+2j-3\hat{k}\right)$ and $2\hat{i}-2\hat{j}+3\hat{k}+\mu \left(-\hat{i}+\hat{j}+2\hat{k}\right)$ respectively.

Find the shortest distance between the metro track and the road, in the model. Show your work.

Find the shortest distance between the lines $\overrightarrow{r}=3\hat{i}+2\hat{j}-4\hat{k}+\lambda (\hat{i}+2\hat{j}+2\hat{k})$and $\overrightarrow{r}=5\hat{i}-2\hat{j}+\mu (3\hat{i}+2\hat{j}+6\hat{k})$.

If the lines intersect find their point of intersection.

Find the shortest distance between the lines $l_1$ and $l_2$ whose vector equations are

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

$\overrightarrow{r}=\widehat{2i}+\hat{j}-\hat{k}+\mu \left(3\hat{i}-\widehat{5j}+2\hat{k}\right)$

Find the shortest distance between the two lines $\frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7}$ and $\frac{x-2}{1}=\frac{y-4}{3}=\frac{z-6}{5}$

The distance between two parallel lines $\overrightarrow{r}=\left(2\hat{i}+5\hat{j}+6\hat{k}\right)+\lambda \left(\hat{i}+\hat{j}\right)$ and $\overrightarrow{r}=\left(3\hat{i}+5\hat{j}+6\hat{k}\right)+\mu \left(\hat{i}+\hat{j}\right)$ is

The shortest distance between the lines $\frac{x+7}{-6}=\frac{y-6}{7}=z$ and $$$\frac{7-x}{2}=y-2=z-6$ is

Find the shortest distance between the lines 

$\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}$ and $\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}$

Find the distance between the lines $l_1$ and $l_2$ given by

$\overrightarrow{r}=\hat{i}+2\hat{j}-4\hat{k}+\lambda \left(2\hat{i}+3\hat{j}+6\hat{k}\right)$

and $\overrightarrow{r}=3\hat{i}+3\hat{j}-5\hat{k}+\mu \left(2\hat{i}+3\hat{j}+6\hat{k}\right)$

Find the shortest distance between the lines $l_1$ and $l_2$ whose vector equations are

$\overrightarrow{r}=\hat{i}+\hat{j}+\lambda \left(2\hat{i}-\hat{j}+\hat{k}\right) ......\left(1\right)$

and $\overrightarrow{r}=2\hat{i}+\hat{j}-\hat{k}+\mu \left(3\hat{i}-5\hat{j}+2\hat{k}\right) ......\left(2\right)$

Find the shortest distance between the following lines:

$\overrightarrow{r}=(\hat{i}+\hat{j}-\hat{k})+s(2\hat{i}+\hat{j}+\hat{k})$

$\overrightarrow{r}=(\hat{i}+\hat{j}+2\hat{k})+t(4\hat{i}+2\hat{j}+2\hat{k})$

If the shortest distance between the lines
$\overrightarrow{r}=(4\hat{i}-\hat{j})+\lambda (\hat{i}+2\hat{j}-3\hat{k})$and $\overrightarrow{r}=(\hat{i}-\hat{j}+2\hat{k})+\mu (2\hat{i}+4\hat{j}-5\hat{k})$ is $\frac{a}{\sqrt{5}}$, then $a=$
 

By computing the shortest distance, determine whether the lines $\overline{r}=(\hat{i}-\hat{j})+\lambda (2\hat{i}+\hat{k})$ and $\overline{r}=2\hat{i}-\hat{j}+\mu (\hat{i}-\hat{j}+\hat{k})$  intersect or not.

If the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$ intersect, then find the value of $k$.

If the shortest distance between the lines $\overrightarrow{r}=(\lambda -1)\hat{i}+(\lambda +1)\hat{j}-(1+\lambda )\hat{k}$ and $\overrightarrow{r}=(1-\mu )\hat{i}+(2\mu -1)\hat{j}+(\mu +2)\hat{k}$ is of the form of $\frac{a}{\sqrt{2}}$, then $a=$ 

Find the shortest distance between the lines given by

$\overline{r}=(8+3\lambda )\hat{i}-(9+16\lambda )\hat{j}+(10+7\lambda )\hat{k}\text{ and }\overline{r}=15\hat{i}+29\hat{j}+5\hat{k}+\mu (3\hat{i}+8\hat{j}-5\hat{k})$.

What is the primary purpose of constructing a line graph?

Which port is the largest container port in India?