• 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?


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.

  • $\begingroup$ To check orthogonality we will take dot product of line and normal, but how does orthogonality inferring that line is parallel. $\endgroup$ – sachal Apr 26 '18 at 15:01
  • $\begingroup$ That's what "normal vector" means: it is orthogonal to the plane, so it is orthogonal to any line in the plane, so it is orthogonal to any line parallel to the plane. $\endgroup$ – Arthur Apr 26 '18 at 15:28

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