The angle between the planes $x+y+2z=6$ and $2x-y+z=9$ is:

$A$ vector $\vec{n}$ is inclined to the $x$-axis at $45^\circ$,to the $y$-axis at $60^\circ$,and at an acute angle to the $z$-axis. If $\vec{n}$ is a normal to a plane passing through the point $(\sqrt{2}, -1, 1)$,then the equation of the plane is:

What is the equation of the plane passing through the point $(1, 1, 0)$ and perpendicular to the line $\vec{r} = (2, 3, 4) + k(3, 4, 5)$,where $k \in R$?

If the foot of the perpendicular drawn from the point $(1,0,-2)$ to the plane $\pi$ is $(2,0,-1)$ and the equation of the plane $\pi$ is $ax+by+cz=2$,then $a^2+b^2+c^2=$

Find the coordinates of the foot of the perpendicular drawn from the origin to the plane $x+y+z=1$.

Let the plane $ax + by + cz = d$ pass through $(2, 3, -5)$ and be perpendicular to the planes $2x + y - 5z = 10$ and $3x + 5y - 7z = 12$. If $a, b, c, d$ are integers,$d > 0$,and $\text{gcd}(|a|, |b|, |c|, d) = 1$,then the value of $a + 7b + c + 20d$ is equal to