If the lines $\frac{x - 5}{5m + 2} = \frac{2 - y}{5} = \frac{1 - z}{-1}$ and $\frac{x}{1} = \frac{2y + 1}{4m} = \frac{1 - z}{-3}$ are perpendicular to each other, then the value of $m$ is ...

Find the vector and the cartesian equations of the line that passes through the points $(3,-2,-5)$ and $(3,-2,6).$

The coordinates of the foot of the perpendicular from the point $(0,2,3)$ on the line $\frac{x+3}{5}=\frac{y+1}{2}=\frac{z+4}{3}$ are

Let the point $A$ be the foot of the perpendicular drawn from the point $P(a, b, 0)$ to the line $\frac{x-1}{2} = \frac{y-2}{1} = \frac{z-\alpha}{3}$. If the midpoint of the line segment $PA$ is $(0, \frac{3}{4}, -\frac{1}{4})$, then the value of $a^2+b^2+\alpha^2$ is equal to:

The angle between the lines $2x = 3y = -z$ and $6x = -y = -4z$ is ......... $^o$.

If the lines $\vec{r} = (\hat{i} + m\hat{j} + 3\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 4\hat{k})$ and $\vec{r} = (4\hat{i} + \hat{j}) + \mu(5\hat{i} + m\hat{j} + \hat{k})$ intersect each other, then $m =$ ?