How to construct a line perpendicular to a plane

Question

I'm well aware of using the cross product to construct a perpendicular to two vectors. Given a plane, is there a way to construct a line perpendicular to it, using just a straightedge and compass? In Book XI Euclid assumes that such a line exists, but did he ever construct it?

Answer 1

I assume we are able to pass planes in 3D space, and on those planes we are able to use ruler and compass and pencil. In that case:

From an arbitrary point not on the given plane, pass 2 planes and let them meet the given plane on two lines.

For each of those lines:

• from the initial arbitrary point, draw the perpendicular on its respective plane.
• Then, from the point where that perpendicular lands on the line, draw another perpendicular to the line, this time on the given plane.

Mark the meeting point of the latter pair of perpendiculars on the given plane. Then connect that point to the initial arbitrary point. The line you just drew is perpendicular to our given plane.