The length of the perpendicular from the point $(0, 2, 3)$ to the line $\frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3}$ is:

Find the acute angle between the line joining the points $(2, 1, -3)$ and $(-3, 1, 7)$ and the line parallel to $\frac{x - 1}{3} = \frac{y}{4} = \frac{z + 3}{5}$ passing through the point $(-1, 0, 4)$.

Let $P(\alpha, \beta, \gamma)$ be the image of the point $Q(3, -3, 1)$ in the line $\frac{x-0}{1} = \frac{y-3}{1} = \frac{z-1}{-1}$ and $R$ be the point $(2, 5, -1)$. If the area of the triangle $PQR$ is $\lambda$ and $\lambda^2 = 14K$,then $K$ is equal to:

If the lines given by $\bar{r} = 2 \hat{i} + \lambda(\hat{i} + 2 \hat{j} + m \hat{k})$ and $\bar{r} = \hat{i} + \mu(2 \hat{i} + \hat{j} + 6 \hat{k})$ are perpendicular,then the value of $m$ is:

If $L_1$ is a line through the point $5 \hat{i}+8 \hat{j}+11 \hat{k}$ and parallel to the vector $2 \hat{i}+3 \hat{j}+4 \hat{k}$ and $L_2$ is a line through the point $4 \hat{i}+6 \hat{j}+8 \hat{k}$ and parallel to the vector $3 \hat{i}+4 \hat{j}+5 \hat{k}$,then the point of intersection of $L_1$ and $L_2$ is

If the harmonic conjugate of $P(2, 3, 4)$ with respect to the line segment joining the points $A(3, -2, 2)$ and $B(6, -17, -4)$ is $Q(\alpha, \beta, \gamma)$,then $\alpha + \beta + \gamma =$