Perpendicular form of the straight line equation in n-dimensional space

Question

A similar question was discussed before but it was in a 2-dimensional space. Perpendicular form of the straight line equation. So how can the derivation in a 2D space be generalized to an n-dimensional space? In other words, how can we derive the perpendicular form of a specific line perpendicular to the hyperplane defined as L = A.X1 + B.X2+ C.X3 + ... + K.Xn in an n-dimensional space?

Answer 1

The vector ${\bf A}=(A_1,A_2, \cdots, A_n)$ is a vector normal to the plane.

Denoting by $\bf X$ the generic point of coordinates $(x_1,\cdots, x_n)$ and by $\bf P$ a given point, then $${\bf(X-P)}=\lambda \bf A$$ is the parametric equation of a line normal to the plane, and passing through $\bf P$.