Angle between Two Lines

Find the angle between the lines $2x=3y=-z$ and $6x=-y=-4z$ is 

The angle between the lines $\widehat{r}=\left(2\widehat{i}-3\widehat{j}+ \widehat{k}\right)+\lambda \left(\widehat{i}+4\widehat{j}+3\widehat{k}\right)$ and $\widehat{r}=\left(\widehat{i}- \widehat{j}+2\widehat{k}\right)+\mu \left(\widehat{i}+2\widehat{j}-3\widehat{k}\right)$ is

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

The angle between the lines whose direction cosines are $\left(\frac{\sqrt{3}}{4},\frac{1}{4},\frac{\sqrt{3}}{2}\right) \mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\left(\frac{\sqrt{3}}{4},\frac{1}{4},\frac{-\sqrt{3}}{2}\right)$ is

Find the acute angle between the following lines.

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

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

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

The angle between the pair of lines $\overrightarrow{r}=3\hat{i}+2\hat{j}-4\hat{k}+\lambda (\hat{i}+2\hat{j}+2\hat{k})$ and $\overrightarrow{r}=5\hat{i}-2\hat{k}+\mu (3\hat{i}+2\hat{j}+6\hat{k})$ is
 

The angle between the lines $\frac{x-1}{4}=\frac{y-3}{1}=\frac{z}{8}$ and $\frac{x-2}{2}=\frac{y+1}{2}=\frac{z-4}{1}$ is

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

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 the straight lines $\frac{x}{1}=\frac{y}{1}=\frac{z}{1}$ and $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ and a third line passing through the point $Q(1,1,1)$ form a triangle whose area is $6^{\frac{1}{2}}$ sq. units, then the point of intersection of second line with third line is:

The direction cosines of two lines are given by $2\mathrm{\ell }+\mathrm{\lambda m}+\mathrm{n}=0$ and $\mathrm{mn}+\mathrm{n\ell }-5\mathrm{\ell m}=0.$ Find value of $\lambda$,if the lines are perpendicular.

What is the condition of perpendicularity of the two lines given as $x=ay+b, z=cy+d$ and $\mathrm{ }x=a^{\prime }y+b^{\prime }, z=c^{\prime }y+d^{\text{'}}$?

Which of the following is true about the two-sector model?

The two lines $x=my+n, z=py+q$ and $x=m^{\prime }y+n^{\prime }, z=p^{\prime }y+q\prime$ are perpendicular to each other, if

The straight lines whose direction cosines are given by $al+bm+cn=0,fmn+gnl+hlm=0$ are perpendicular, if

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