Interaction between Two Lines in 3D

If the lines $\frac{x-1}{-3}=\frac{y-2}{2k}=\frac{z-3}{2}$ and $\frac{x-1}{3k}=y-1=\frac{z-6}{-5}$ are perpendicular, then the value of $k$ is

Two lines given by $\overrightarrow{r}=4\hat{i}-5\hat{j}+\hat{k}+\lambda \left(2\hat{i}+4\hat{j}+3\hat{k}\right)$ and $\overrightarrow{r}=2\hat{i}-\hat{j}+\mu \left(\hat{i}+3\hat{j}+2\hat{k}\right)$ are

Let two lines $L_1$ and $L_2$ be given by the vector equations $\overrightarrow{r}=\hat{i}+\hat{j}+\lambda (2\hat{i}-\hat{j}+\hat{k})$ and $\overrightarrow{r}=2\hat{i}+\hat{j}-\hat{k}+\mu (3\hat{i}-5\hat{j}+2\hat{k})$ respectively. The shortest distance between $L_1$ and $L_2$ is

Let $L_1,L_2$ and $L_3$ be lines given by

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

Then,

The shortest distance between the lines given by

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

is equal to

If $L_1$ is a line passing through the origin, and having $2,2,1$ as the direction ratios, and if $L_2$ is a line passing through $(5,2,3)$, and having $4,1,8$ as the direction ratios, then the cosine of the angle between $L_1$ and $L_2$ is

Let $l_0$ be the line defined by the vector (equation) $\hat{i}+2\hat{j}+3\hat{k}+\lambda (\hat{i}+\hat{j}+\hat{k})$, with $\lambda$ real. Which of the following vector equations, with $\mu$ real, defines a line which intersects $l_0$ ?

The value of $\alpha$ for which the shortest distance between the lines represented by $y+z=0, z+x=0$ and $x+y=0, x+y+z=\alpha$ is $1,$ is

The set of all non-zero real values of $k$, for which the lines $\frac{x-4}{2}=\frac{y-6}{2}=\frac{z-8}{-2k^2}$ and $\frac{x-2}{2k^2}=\frac{y-8}{4}=\frac{z-10}{2}$ are coplanar

If the shortest distance between the lines $x+2\lambda =2y=-12z, x=y+4\lambda =6z-12\lambda$ is $4\sqrt{2}$ units, then value of $\lambda$ is

Find the acute angle between the following lines.

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

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

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

The acute angle between the lines $\overrightarrow{r}=\left(4\hat{i}-\hat{j}\right)+t\left(\hat{i}+2\hat{j}-2\hat{k}\right)$ and$\overrightarrow{r}=\left(\hat{i}-2\hat{j}+4\hat{k}\right)+s\left(-\hat{i}-2\hat{j}+2\hat{k}\right)$ is