$A$ line with direction cosines passes through the point $P(2, -1, 2)$ and makes equal angles with the coordinate axes. The line meets the plane $2x + y + z = 9$ at point $Q$. Find the length of the line segment $PQ$.

If the lines $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4}$ and $\frac{x - 3}{1} = \frac{y - k}{2} = \frac{z}{1}$ intersect,then $k$ is equal to:

The image of the point $(5, 2, 6)$ with respect to the plane $x + y + z = 9$ is

Find the distance of the point $2\hat{i} + \hat{j} - \hat{k}$ from the plane $\vec{r} \cdot (\hat{i} - 2\hat{j} + 4\hat{k}) = 9$.

Let $\bar{A}$ be a vector parallel to the line of intersection of planes $P_1$ and $P_2$ passing through the origin. $P_1$ is parallel to the vectors $2 \hat{j}+3 \hat{k}$ and $4 \hat{j}-3 \hat{k}$,and $P_2$ is parallel to $\hat{j}-\hat{k}$ and $3 \hat{i}+3 \hat{j}$. Then the angle between $\bar{A}$ and $2 \hat{i}+\hat{j}-2 \hat{k}$ is

For what value of $c$ does the line $\frac{x - 2}{3} = \frac{y + 1}{2} = \frac{z - 1}{-1}$ intersect the curve $xy = c^2, z = 0$?