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Perpendicularity between lines and planes

Perpendicularity between a line and a plane follows a very concrete rule in dihedral: the projections of the line are perpendicular to the matching traces of the plane. That is, r′ is perpendicular to α₁ and r″ is perpendicular to α₂. That rule solves both the perpendicular through a point and the plane perpendicular to a line.

What you need to know

• Draw the line perpendicular to a plane through a given point.
• Draw the plane perpendicular to a line through a point.
• Recognise when two lines are perpendicular even if they do not look 90° in any projection.
• Use the theorem of the three perpendiculars in skew-line cases.

How it is solved

1. Line perpendicular to a plane: through the point draw r′ perpendicular to α₁ and r″ perpendicular to α₂.
2. Plane perpendicular to a line: make its traces perpendicular to the projections of the line.
3. If the plane is projecting, the perpendicular is a pointing or standing line: check it in the matching projection.
4. For two intersecting lines, rotate the plane that contains them flat and check the 90° angle there.

How it appears in the exam

They ask for “line perpendicular to plane α through P” and, the other way round, “plane perpendicular to r through P”. It also appears as a previous step for distances: the perpendicular is what lets you find the foot.

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