If the angle between the planes $x-2y+3z-5=0$ and $x+\alpha y+2z+7=0$ is $\cos^{-1}\left(\frac{1}{14}\right)$,then the difference between the values of $\alpha$ 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 .... .

If $a, b, c$ are the intercepts made by the plane passing through the point $(1, 2, 3)$ parallel to the plane $3x + 4y - 5z = 0$ on the $X, Y, Z$-axes respectively,then $3a + b + 5c =$

The direction ratios of the two lines $AB$ and $AC$ are $1, -1, -1$ and $2, -1, 1$. The direction ratios of the normal to the plane $ABC$ are

If the distance between the plane,$23x - 10y - 2z + 48 = 0$ and the plane containing the lines $\frac{x+1}{2} = \frac{y-3}{4} = \frac{z+1}{3}$ and $\frac{x+3}{2} = \frac{y+2}{6} = \frac{z-1}{\lambda}$ $(\lambda \in R)$ is equal to $\frac{k}{\sqrt{633}}$,then $k$ is equal to

$A$ plane $E_{1}$ makes intercepts $1, -3, 4$ on the coordinate axes. The equation of a plane parallel to plane $E_{1}$ and passing through $(2, 6, -8)$ is