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 = $

The coordinates of the foot of the perpendicular drawn from the origin to the plane $2x - y + 5z - 3 = 0$ are $ . . . . . . $.

Find the vector and Cartesian equations of the plane that passes through the point $(1, 0, -2)$ and the normal to the plane is $\hat{i} + \hat{j} - \hat{k}$.

The Cartesian equation of the plane passing through the point $A(7,8,6)$ and parallel to the $XY$ plane is

If the angle between the planes $\bar{r} \cdot(11 \hat{i}-2 \hat{j}+\alpha \hat{k})=7$ and $\bar{r} \cdot(2 \hat{i}+4 \hat{j}-2 \hat{k})=5$ is $\frac{\pi}{2}$,then $\alpha=$

The direction ratios of the normal to the plane passing through $(1, 0, 0)$ and $(0, 1, 0)$ which makes an angle $\frac{\pi}{4}$ with the plane $x + y = 3$ are: