The lines $x = ay + b, z = cy + d$ and $x = a'y + b', z = c'y + d'$ are perpendicular to each other,if

Find the length of the perpendicular drawn from the point $P(3, -1, 11)$ to the line $\frac{x}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$.

If the coordinates of the points $A, B, C,$ and $D$ are $(1, 2, 3), (4, 5, 7), (-4, 3, -6),$ and $(2, 9, 2)$ respectively,then find the angle between the lines $AB$ and $CD$. (in $^\circ$)

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

Let a triangle $PQR$ be such that $P$ and $Q$ lie on the line $\frac{x+3}{8} = \frac{y-4}{2} = \frac{z+1}{2}$ and are at a distance of $6$ units from $R(1, 2, 3)$. If $(\alpha, \beta, \gamma)$ is the centroid of $\triangle PQR$,then $\alpha + \beta + \gamma$ is equal to :

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