Determine if two planes are intersecting in parametric form

Question

given two planes in parametric form :

$\pi_1:(x_1,y_1,z_1)+t\overrightarrow{u_1}+k\overrightarrow{v_1}$

$\pi_2:(x_2,y_2,z_2)+s\overrightarrow{u_2}+p\overrightarrow{v_2}$

How can i determine if the two planes are intersecting without converting to algebraic forms?

Is it possible?

thanks.

Answer 1

One way: $\pi_1$ and $\pi_2$ intersect iff the system of linear equations on the variables $t,k,s,p $ $$P_1+t\overrightarrow{u_1}+k\overrightarrow{v_1}=P_2+s\overrightarrow{u_2}+p\overrightarrow{v_2}$$ is consistent. Equivalently, iff $$\text{rank }[\overrightarrow{u_1},\overrightarrow{v_1},-\overrightarrow{u_2},-\overrightarrow{v_2}]=\text{rank }[\overrightarrow{u_1},\overrightarrow{v_1},-\overrightarrow{u_2},-\overrightarrow{v_2},P_2-P_1]$$ (Vectors and points in columns).