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 $\alpha=$

Find the equation of the plane passing through the point $(\alpha, \beta, \gamma)$ and parallel to the plane $x + y + z = 0$.

The equation of the plane in Cartesian form, which is at a distance of $\frac{6}{\sqrt{29}}$ from the origin and its normal vector drawn from the origin being $2 \hat{i}-3 \hat{j}+4 \hat{k}$, is

The equation of the plane bisecting the line segment joining the points $(2,0,6)$ and $(-6,2,4)$ and perpendicular to it,is

$A$ plane passes through the points $A (1, 2, 3)$,$B (2, 3, 1)$,and $C (2, 4, 2)$. If $O$ is the origin and $P$ is $(2, -1, 1)$,then the projection of $\overline{OP}$ on this plane is of length .... .

The plane passing through the points $(1,1,1)$,$(1,-1,1)$ and $(-7,-3,-5)$ is