平面方程与距离问题

A plane is fixed by one point and one normal direction

Three points determine a plane through two cross products

Distance from a point to a plane divides by the length of the normal

Distance from a point to a line uses the cross product

A plane passes through $P_0(1, 2, 3)$ and has normal vector $\mathbf{n} = (2, -1, 4)$. Which equation describes the plane?

A plane passes through $P(1, 0, 0)$, $Q(0, 2, 0)$ and $R(0, 0, 3)$. Which vector is normal to the plane?

How far is the point $Q(3, 1, 2)$ from the plane $2x - y + 4z = 12$?

The planes $2x - y + 4z = 12$ and $2x - y + 4z = 2$ are parallel. What is the distance between them?

A line passes through $(1, 1, 0)$ with direction $\mathbf{v} = (0, 1, 2)$, and $Q(3, 1, -2)$ is off the line. Give the exact distance, written in the form (a*sqrt(b))/c with integers, like 2*sqrt(3)/5.

A line is parametrized by $(x, y, z) = (1, 2, 0) + t(1, 0, 1)$ and a plane has equation $x + y + z = 6$. Give the value of the parameter $t$ at the point where the line crosses the plane, as a fraction like 3/2.