Math formulas and cheat sheets for Planes in three dimensions

Planes in three dimensions

Plane forms

Point direction form:

 a(x−x1)+b(y−y1)+c(z−z1)=0

where P(x1,y1,z1)$P(x_1, y_1, z_1)$ lies in the plane, and the direction (a,b,c)$(a,b,c)$ is normal to the plane.

General form:

 Ax+By+Cz+D=0

where direction (A,B,C)$(A,B,C)$ is normal to the plane.

Intercept form:

 xa+yb+zc=1

this plane passes through the points (a,0,0),(0,b,0)$(a,0,0), (0,b,0)$ and (0,0,c)$(0,0,c)$.

Three point form:

 ∣∣∣∣x−x3x1−x3x2−x3y−y3y1−y3y2−y3z−z3z1−z3z2−z3∣∣∣∣=0

Normal form:

 xcosα+ycosβ+zcosγ=p

Parametric form:

 xyz=x1+a1s+a2t=y1+b1s+b2t=z1+c1s+c2t

where the directions (a1,b1,c1)$(a_1, b_1, c_1)$ and (a2,b2,c2)$(a_2, b_2, c_2)$ are parallel to the plane.

Angle between two planes:

The angle between planes A1x+B1y+C1z+D1=0$A_1x + B_1y + C_1z + D_1 = 0$ and A2x+B2y+C2z+D2=0$A_2x + B_2y + C_2z + D_2 = 0$ is:

 α=arccosA1A2+B1B2+C1C2A21+B21+C21−−−−−−−−−−−√⋅A22+B22+C22−−−−−−−−−−−√

The planes are parallel if and only if

 A1A2=B1B2=C1C2

Equation of a plane

The equation of a plane through P1(x1,y1,z1)$P_1(x_1, y_1, z_1)$ and parallel to directions (a1,b1,c1)$(a_1, b_1, c_1)$ and (a2,b2,c2)$(a_2, b_2, c_2)$ has an equation:

 ∣∣∣∣x−x1a1a2y−y1b1b2z−z1c1c2∣∣∣∣=0

The equation of a plane through P1(x1,y1,z1)$P_1(x_1, y_1, z_1)$ andP1(x2,y2,z2)$P_1(x_2, y_2, z_2)$), and parallel to direction (a,b,c)$(a,b,c)$, has equation

 ∣∣∣∣x−x1x2−x1ay−y1y2−y1bz−z1z2−z1c∣∣∣∣=0

The equation of a plane through P1(x1,y1,z1)$P_1(x_1, y_1, z_1)$ , P2(x2,y2,z2)$P_2(x_2, y_2, z_2)$ and P3(x3,y3,z3)$P_3(x_3, y_3, z_3)$ , has equation

 ∣∣∣∣x−x1x2−x1x3−x1y−y1y2−y1y3−y1z−z1z2−z1z3−z1∣∣∣∣=0

Distance from point to plane

The distance of P1(x1,y1,z1)$P_1(x_1, y_1, z_1)$ from the plane Ax+By+Cz+D=0$Ax + By + Cz + D = 0$ is

 d=Ax1+By1+Cz1A2+B2+C2−−−−−−−−−−−√

Intersection of two planes

The intersection of planes A1x+B1y+C1z+D1=0$A_1x + B_1y + C_1z + D_1 = 0$ and A2x+B2y+C2z+D2=0$A_2x + B_2y + C_2z + D_2 = 0$ is the line:

 x−x1a=y−y1b=z−z1c

where

 ax1y1z1=∣∣∣B1B2C1C2∣∣∣  b=∣∣∣C1C2A1A2∣∣∣  c=∣∣∣A1A2B1B2∣∣∣=b∣∣∣D1D2C1C2∣∣∣−c∣∣∣D1D2B1B2∣∣∣a2+b2+c2=c∣∣∣D1D2A1A2∣∣∣−a∣∣∣D1D2C1C2∣∣∣a2+b2+c2=a∣∣∣D1D2B1B2∣∣∣−b∣∣∣D1D2A1A2∣∣∣a2+b2+c2

If a=b=c=0$a = b = c = 0$, then the planes are parallel.