# Perpendicular line to another line locked to a plane

I have a line, A, formed of a vector and a point, which is on a plane, P.

How would I go about finding the line, B, that runs perpendicular to A while also being contained within the plane P?

(I have two such vectors on the same plane, and need to find the intersection line between the two perpendicular lines from the two).

• What did you try? – Peter Franek Aug 7 '14 at 13:54
• try finding a vector (N) normal to the plane (P). and calculate $$N\times A$$ – John Joy Aug 7 '14 at 13:57

I'm assuming you're working in three dimensions.

Hint #1: The set of lines perpendicular to $A$ form a plane.

Then the line you seek is the intersection of this plane with your given plane $P$.

An equivalent approach:

Hint #2: Project $A$ onto $P$; now the problem is in two dimensions (the plane $P$).

Then find the line perpendicular to $\operatorname{Proj}(A)$ in $P$; this line in the original 3-dimensional space will still be perpendicular to $A$.

Say the line $A$ goes along the vector $\vec{a}$. The plane has a normal vector $\vec{p}$. The cross product $\vec{p}\times\vec{a}$ gives you a vector that is perpendicular to $\vec{p}$ (hence in the plane) and also perpendicular to $\vec{a}$ (hence perpendicular to your line).

• Aha, that makes sense. I'll give it a go now. Thanks! – Incredidave Aug 7 '14 at 13:59