AHL 3.17 (HL)—Planes

A plane is fixed by a point and a normal vector.

A normal n is perpendicular to every direction in the plane, so n·(r−a)=0 gives a compact equation for the plane through a.

Example

Through (1,0,2) with normal (2,1,−1), the plane is 2(x−1)+y−(z−2)=0.

Substitute the known point to determine the constant, then test any proposed point with the dot product equation.

A direction vector lying in the plane is perpendicular to the normal; it is not itself the plane’s normal.

A plane through point vector $\mathbf a$ can be written $\mathbf r=\mathbf a+\lambda\mathbf b+\mu\mathbf c$ using two non-parallel directions in the plane, or $\mathbf r\cdot\mathbf n=\mathbf a\cdot\mathbf n$ using a normal $\mathbf n$. If $\mathbf n=(A,B,C)$, this becomes $Ax+By+Cz=d$. To convert from two in-plane directions to normal form, use $\mathbf n=\mathbf b\times\mathbf c$.