The equation of the line passing through the points $(3, 2, 4)$ and $(4, 5, 2)$ is

Find the equation of the line passing through the point $(1, 2, -4)$ and perpendicular to the two lines $\frac{x - 8}{3} = \frac{y + 19}{-16} = \frac{z - 10}{7}$ and $\frac{x - 15}{3} = \frac{y - 29}{8} = \frac{z - 5}{-5}$.

Let $A$ and $B$ be two points on the line $\frac{x}{1} = \frac{y}{1} = \frac{z}{-1}$. If the distance of point $P(1, 1, 1)$ from the points $A$ and $B$ is $\sqrt{3}$,then the distance between $A$ and $B$ is:

The graph of the equation $y^2 + z^2 = 0$ in three-dimensional space is

The point of intersection of the lines represented by $r=(\hat{i}+2 \hat{j}-\hat{k})+\lambda(2 \hat{i}+3 \hat{j}+4 \hat{k})$ and $r=(-\hat{i}-3 \hat{j}+7 \hat{k})+\mu(\hat{i}+2 \hat{j}-\hat{k})$ is

Statement $1:$ The shortest distance between the lines $\frac{x}{2} = \frac{y}{-1} = \frac{z}{2}$ and $\frac{x-1}{4} = \frac{y-1}{-2} = \frac{z-1}{4}$ is $\sqrt{2}$.
Statement $2:$ The shortest distance between two parallel lines is the perpendicular distance from any point on one of the lines to the other line.