The equation of the plane passing through the points $(1, 2, -3)$ and $(2, -2, 1)$ and parallel to the $X$-axis is:

The equation of the plane passing through the point $(1, 2, -3)$ and perpendicular to the planes $3x + y - 2z = 5$ and $2x - 5y - z = 7$ is:

The position vectors of two points $P$ and $Q$ are $3i + j + 2k$ and $i - 2j - 4k$ respectively. The equation of the plane passing through $Q$ and perpendicular to $PQ$ is:

If $\alpha$ is the acute angle between the planes $P_1$ and $P_2$,where the combined equation of the planes $P_1$ and $P_2$ is $2x^2 - 6y^2 - 12z^2 + 18yz + 2zx + xy = 0$,then the value of $\cos \alpha$ is:

Find the coordinates of the foot of the perpendicular drawn from the origin to the plane $5y + 8 = 0$.

If the foot of the perpendicular from $(0,0,0)$ to a plane is $(1,2,3)$,then the equation of the plane is