3D Geometry / HL

Angle Between Two Planes

The angle between two planes is found using the angle between their normal vectors.

Formula

If planes have normal vectors \(\mathbf n_1\) and \(\mathbf n_2\), then

\[\cos\theta=\frac{|\mathbf n_1\cdot\mathbf n_2|}{|\mathbf n_1||\mathbf n_2|}.\]

The absolute value gives the acute angle between the planes.

nP(x,y,z)plane
A normal vector is perpendicular to every direction lying in the plane.

Worked example

For \(x+2y+2z=5\) and \(2x-y+2z=3\), use \(\mathbf n_1=(1,2,2)\) and \(\mathbf n_2=(2,-1,2)\).

\[\mathbf n_1\cdot\mathbf n_2=2-2+4=4.\]