Let the line $\frac{x-3}{7}=\frac{y-2}{-1}=\frac{z-3}{-4}$ intersect the plane containing the lines $\frac{x-4}{1}=\frac{y+1}{-2}=\frac{z}{1}$ and $4 a x-y+5 z-7 a=0=2 x-5 y-z-3, a \in \mathbb{R}$ at the point $P(\alpha, \beta, \gamma)$. Then the value of $\alpha+\beta+\gamma$ equals _____________.

If $\lambda_{1} < \lambda_{2}$ are two values of such that the angle between the planes and is , then the square of the length of perpendicular from the point to the plane is ______________.

Let the line $l: x=\frac{1-y}{-2}=\frac{z-3}{\lambda}, \lambda \in \mathbb{R}$ meet the plane at the point . If the angle between the line and the plane is , then is equal to ___________.

Let $\mathrm{P}_{1}$ be the plane $3 x-y-7 z=11$ and be the plane passing through the points , and . If the foot of the perpendicular drawn from the point on the line of intersection of the planes and is , then is equal to ___________.

Let d be the distance between the foot of perpendiculars of the points P(1, 2, $-$1) and Q(2, $-$1, 3) on the plane $-$x + y + z = 1. Then d is equal to ___________.

Let $\lambda_{1}, \lambda_{2}$ be the values of for which the points and are at equal distance from the plane . If , then the distance of the point from the line is ____________.

Let the equation of the plane P containing the line $x+10=\frac{8-y}{2}=z$ be and the distance of the plane from the point (1, 27, 7) be . Then is equal to __________.

Let the line $L: \frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-3}{1}$ intersect the plane at the point . Let the point be the foot of perpendicular from the point on the line . If is the area of triangle , then is equal to __________.

Let a line with direction ratios $a,-4 a,-7$ be perpendicular to the lines with direction ratios $3,-1,2 b$ and . If the point of intersection of the line and the plane is , then is equal to _________.

Let $\alpha x+\beta y+\gamma z=1$ be the equation of a plane passing through the point and perpendicular to the line joining the points and . Then the value of is equal to _____________.

The plane passing through the line $L: l x-y+3(1-l) z=1, x+2 y-z=2$ and perpendicular to the plane is . If is the acute angle between the line and the -axis, then is equal to _____________.

Let the lines

${L_1}:\overrightarrow r = \lambda \left( {\widehat i + 2\widehat j + 3\widehat k} \right),\,\lambda \in R$