Angle Between Planes

Topics: Geometry - 3D Space - Plane - Angle

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(theorem)

The angle $\theta$ between two planes ${\Pi }_1$ and ${\Pi }_2$ with their respective equations…

$${\Pi }_1:Ax+B1y+Cz+D=0$$

$${\Pi }_2:A^{\prime }x+B^{\prime }y+C^{\prime }z+D^{\prime }=0$$

…is determined by the equality:

$$\cos \theta =\frac{|A{A}^{\prime }+B{B}^{\prime }+C{C}^{\prime }|}{\sqrt{A^2+B^2+C^2}\sqrt{(A^{\prime }{)}^2+(B^{\prime }{)}^2+(C^{\prime }{)}^2}}$$

Given the respective normal vectors, we can rewrite this equality as:

$$\cos \theta =\frac{|n_1\cdot n_2|}{\Vert n_1\Vert \Vert n_2\Vert }$$

Notice the formula is very similar to the one used to obtain the angle between lines.

Criterium §

• The planes ${\Pi }_1$ and ${\Pi }_2$ are parallel if and only if there exists a real number $k\ne 0$ such that:

$$A=k{A}^{\prime },B=k{B}^{\prime },C=k{C}^{\prime }$$

• The planes ${\Pi }_1$ and ${\Pi }_2$ are the same plane if they’re parallel and if $D=k{D}^{\prime }$.

• The planes ${\Pi }_1$ and ${\Pi }_2$ are orthogonal if and only if $A{A}^{\prime }+B{B}^{\prime }+C{C}^{\prime }=0$ (i.e. if $n_1\cdot n_2=0$).