Let each plane be defined by a point (pₖ) and a unit normal vector (vₖ), (k = 1, 2, 3). Then the unique point of intersection can be written in closed form as:
a) (p₁ {(p₂ - p₁) ⋅ (v₂ × v₃)}/{v₁ ⋅ (v₂ × v₃)} ⋅ v₁)
b) (p₁ - {(p₂ - p₁) ⋅ (v₂ × v₃)}/{v₁ ⋅ (v₂ × v₃)} ⋅ v₁)
c) (p₁ {(p₂ - p₁) ⋅ (v₂ × v₃)}/{v₁ ⋅ (v₂ × v₃)} ⋅ v₂)
d) (p₁ - {(p₂ - p₁) ⋅ (v₂ × v₃)}/{v₁ ⋅ (v₂ × v₃)} ⋅ v₂)