If the plane passing through the points $(1, 2, 3)$, $(2, 3, 1)$, and $(3, 1, 2)$ is $a x + b y + c z = 1$, then $a + 2 b + 3 c = $

In the following case,determine the direction cosines of the normal to the plane and the distance from the origin: $x+y+z=1$

If a plane meets the coordinate axes at $A, B$ and $C$ in such a way that the centroid of $\triangle ABC$ is at the point $(1, 2, 3)$,then the equation of the plane is

$A$ plane $\pi$ is passing through $(2,0,1)$ and $(3,-3,4)$ and is perpendicular to the plane $x-2y+z=6$. Among the following,the plane that is perpendicular to the plane $\pi$ is:

Find the equation of the plane passing through the origin and parallel to the plane $3x - 4y + 5z - 6 = 0$.

The vector equation of a plane which is at a distance of $5 \text{ units}$ from the origin and normal to the vector $\vec{n} = 2\hat{i} + \hat{j} - 2\hat{k}$ is: