Check if line lies between two planes

Question

I have a line given by an equation $$vec{p} = vec{l}_0 + dvec{l}$$ where $vec{l}$ is a direction vector of a line, $l_0$ is a point on a line and $d$  is some scalar. I know that this line is parallel to two planes, both given by a point lying on a plane and a normal. I want to distinguish if the line lies between those two planes.

Intelligenti pauca · Accepted Answer

Just consider the projections on a line perpendicular to the planes.
Let $vec a$, $vec b$ be the points on the planes, and $vec n$ the common normal vector of the planes (I'm assuming the planes are parallel, otherwise the question is meaningless). The line lies between the planes if
$$
vec acdotvec n < vec l_0cdotvec n < vec bcdotvec n
$$
or the analogous reversed inequality.

Check if line lies between two planes

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