Why line passing through origin in the intersection of three planes is parallel to the planes?

Question

• There are three planes $P_1, P_2$ and $P_3$ and they all pass through origin so it will have infinite solutions.

• The normal to these planes $n_1, n_2$ and $n_3$ are coplanar.

• The line passing though origin is perpendicular to normal vectors but is parallel to three planes.

Why is that line passing through origin parallel to the three planes?

Answer 1

A line is parallel to a plane iff it is orthogonal to the normal vector of that plane. The three normal vectors are coplanar, which means that there is a line which is simultaneously orthogonal to all of them. That means this line is simultaneously parallel to each of the planes.