equations of planes
vector equation
$$ E: \vec{x} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} + s \begin{pmatrix} u_1 \\ u_2 \\ u_3 \end{pmatrix} + t \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}$$
normal equation
$$ E: [ \vec{x} - \vec{a} ] \cdot \vec{n} = 0$$
Cartesian equation
$$ E: n_1 x + n_2 y + n_3 z = d$$
where $ \vec{n} = \begin{pmatrix} n_1 \\ n_2 \\ n_3 \end{pmatrix}$
is the normal vector to the plane E.