If the shortest distance between the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{\lambda}$ and $\frac{x-2}{1}=\frac{y-4}{4}=\frac{z-5}{5}$ is $\frac{1}{\sqrt{3}}$,then the sum of possible values of $\lambda$ is

Statement $-1$: The point $A(1, 0, 7)$ is the mirror image of the point $B(1, 6, 3)$ in the line $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$.
Statement $-2$: The line $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$ bisects the line segment joining $A(1, 0, 7)$ and $B(1, 6, 3)$.

The angle between the lines $3x + 2y + z = 0 = x + y - 2z$ and $2x - y - z = 0 = 7x + 10y - 8z$ is

If $A(3,4,5)$,$B(4,6,3)$,$C(-1,2,4)$,and $D(1,0,5)$ are points such that the angle between the lines $DC$ and $AB$ is $\theta$,then $\cos \theta$ is equal to

The lines $\overrightarrow{r} = (\hat{i} - \hat{j}) + \ell(2\hat{i} + \hat{k})$ and $\overrightarrow{r} = (2\hat{i} - \hat{j}) + m(\hat{i} + \hat{j} - \hat{k})$:

The angle between the lines $\frac{x-1}{l}=\frac{y+1}{m}=\frac{z}{n}$ and $\frac{x+1}{m}=\frac{y-3}{n}=\frac{z-1}{l}$,where $l > m > n$ and $l, m, n$ are roots of the equation $x^3+x^2-4x-4=0$,is