Relative positions: lines and planes

Lines and planes in space

In space, two lines can be: secant (they intersect at a single point), parallel (pointing in the same direction, never intersecting) or non-coplanar. This last situation does not exist in a plane: two lines in three-dimensional space may never intersect without being parallel, as they do not lie in the same plane.

A practical example: in a cubic room, the edge of the ceiling at the front and the vertical edge of the left-hand wall do not intersect and are not parallel: they are non-coplanar.

Line and plane

A line and a plane can be:

• intersecting: they intersect at a single point
• parallel: the line never intersects the plane (or is contained within the plane)

Two planes

Two distinct planes in space are either parallel (no points in common) or intersecting: their intersection is then a line.

Orthogonality

A line is orthogonal to a plane if it is perpendicular to all the lines in that plane passing through their point of intersection. In practice, it is sufficient to check that it is perpendicular to two intersecting lines in the plane to draw this conclusion.

Situation	Possible intersection
Two lines	a point, none (parallel or not coplanar)
Line and plane	a point, none, or the line lies within the plane
Two planes	none, or a line

Common pitfall

A common mistake is to assume that two lines which never intersect must necessarily be parallel: in three-dimensional space, this is only true if they lie in the same plane. Always check whether the lines lie in the same plane before concluding that they are parallel.