Characterization of a plane in the space

α\alpha α is a plane in the space and we've been told in school that its equation in canonical form is: ax+by+cz+d=0 ax+by+cz+d=0 a x + b y + cz + d = 0 But this doesn't tell us nothing about the plane characteristics and no-one explained us how this equation is formed. Plane formation A plane in the space could be identified by a point and 2 non-parallel vectors. P=(x0,y0,z0)∈αu‾,v‾∈α P=(x_0,y_0,z_0)\in \alpha \\ \\ \underline{u},\underline{v}\in \alpha P = ( x 0 ​ , y 0 ​ , z 0 ​ ) ∈ α u ​ , v ​ ∈ α u‾,v‾\underline{u},\underline{v} u ​ , v ​ have magnitude, direction, and orientation. It's possible to write: u‾=PQv‾=PR \underline{u}=PQ \\ \underline{v}=PR u ​ = PQ v ​ = PR Every other point XX X that belongs to α\alpha α could be written as a linear combination of these elements starting from the origin: OX=OP+s∗u‾+t∗v‾X=(x,y,z)∈αs∈Rt∈R OX=OP+s*\underline{u}+t*\underline{v} \\ X=(x,y,z)\in\alpha \\ s\in R \\ t\in R OX = OP + s ∗ u ​ + t ∗ v ​ X = ( x , y , z ) ∈ α s ∈ R t ∈ R This cl