Interaction between Two Lines in 3D

State whether the following lines are parallel or not.

$\overrightarrow{r}=\left(4\hat{i}-3\hat{j}+\hat{k}\right)+\lambda \left(\hat{i}-4\hat{j}+7\hat{k}\right)$ and $\overrightarrow{r}=\left(\hat{i}-2\hat{j}+2\hat{k}\right)+\mu \left(\hat{i}-\hat{j}+4\hat{k}\right)$

State whether the following lines are parallel or not.

$\overrightarrow{r}=\left(4\hat{i}-3\hat{j}+\hat{k}\right)+\lambda \left(\hat{i}-4\hat{j}+7\hat{k}\right)$ and $\overrightarrow{r}=\left(2\hat{i}-\hat{j}+\hat{k}\right)+\mu \left(2\hat{i}-8\hat{j}+14\hat{k}\right)$

State whether following vectors are parallel or not.

$\overrightarrow{a}=\hat{i}+2\hat{j}+3\hat{k} \& \overrightarrow{b}=2\hat{i}+4\hat{j}+6\hat{k}$

State whether following vectors are parallel or not.

$\overrightarrow{a}=2\hat{i}+4\hat{j}+5\hat{k} \& \overrightarrow{b}=4\hat{i}-8\hat{j}+10\hat{k}$

Find the shortest distance between the given lines.

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

$\overrightarrow{r}=(-9\hat{i}+\hat{j}+6\hat{k})+\mu (4\hat{i}+\hat{j}+6\hat{k})$

Find the shortest distance between the given lines.

$\overrightarrow{r}=(3-t)\hat{i}+(4+2t)\hat{j}+(t-2)\hat{k}$

$\overrightarrow{r}=(1+s)\hat{i}+(3s-7)\hat{j}+(2s-2)\hat{k}$

The measure of acute angle between the lines whose direction ratios are $3,2,6$ and $-2,1,2$ is

If the lines $\frac{x-1}{5}=\frac{y+1}{3}=\frac{3-z}{\lambda }$ and $\frac{x+1}{4}=\frac{1-3y}{15}=z+1$ are perpendicular to each other, then $\lambda =$

The angle between the lines $\overrightarrow{r}=\left(\hat{ı}+2\hat{ȷ}+3\hat{k}\right)+\lambda \left(\hat{ı}+\hat{ȷ}+2\hat{k}\right)$ and $\overrightarrow{r}=\left(3\hat{ı}+\hat{k}\right)+{\lambda }^{\text{'}}\left(2\hat{ı}+\hat{ȷ}-\hat{k}\right), \lambda , {\lambda }^{\text{'}}\in R$ is

If the lines given by $\frac{x-1}{2\lambda }=\frac{y-1}{-5}=\frac{z-1}{2}$ and $\frac{x+2}{\lambda }=\frac{y+3}{\lambda }=\frac{z+5}{1}$ are parallel, then the value of $\lambda$ is

Given $\Delta ABC$, such that $A$ is $2\hat{i}-\hat{j}+\hat{k}, B$ is $\hat{i}-3\hat{j}-5\hat{k}$ and $C$ is $3\hat{i}-4\hat{j}-4\hat{k}$ then $\Delta ABC$ is

If two lines with equations $\frac{\mathrm{x}-3}{2}=\frac{\mathrm{y}-2}{3}=\frac{\mathrm{z}-1}{\mathrm{\lambda }}$ and $\frac{\mathrm{x}-2}{3}=\frac{\mathrm{y}-3}{2}=\frac{\mathrm{z}-2}{3}$ lie in the same plane, then the value of ${\sin }^{-1}\sin \lambda$ is

Distance between the point $(1, -4, 7)$ and the point of intersection of the lines $\frac{x-4}{1}=\frac{y+3}{-4}=\frac{z+1}{7} \& \frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+10}{8}$ is

The lines $\frac{x-a+d}{\mathrm{\alpha }-\mathrm{\delta }}=\frac{y-a}{\mathrm{\alpha }}=\frac{z-a-d}{\mathrm{\alpha }+\mathrm{\delta }}$ and $\frac{x-b+c}{\mathrm{\beta }-\mathrm{\gamma }}=\frac{y-b}{\mathrm{\beta }}=\frac{z-b-c}{\mathrm{\beta }+\mathrm{\gamma }}$ are coplanar and then equation to the plane in which they lie, is

If three mutually perpendicular lines have direction cosines $\left(l_1,m_1,n_1\right),\left(l_2,m_2,n_2\right)$ and $\left(l_3,m_3,n_3\right)$ , then the line having direction cosines $l_1+l_2+l_3,m_1+m_2+m_3$ and $n_1+n_2+n_3$ make an angle of .... with each other

A point moves so that the sum of its distances from the points $\left(4,\mathrm{ }0,\mathrm{ }0\right)$ and $\left(-4,\mathrm{ }0,\mathrm{ }0\right)$ remains 10. The locus of the point is

A plane which passes through the point $\left(3,\mathrm{ }2,\mathrm{ }0\right)$ and the line $\frac{x-3}{1}=\frac{y-6}{5}=\frac{z-4}{4}$ is

The angle between the line $\frac{x-1}{2}=\frac{y-2}{1}=\frac{z+3}{-2}$ and the plane $x+y+4=0$ , is

The equation of the plane passing through the lines $\frac{x-4}{1}=\frac{y-3}{1}=\frac{z-2}{2}$ and $\frac{x-3}{1}=\frac{y-2}{-4}=\frac{z}{5}$ is