A single linear equation can be used to solve for one variable in terms of the others. It reduces the dimension of the set of its solutions by 1.
Thus the points in a plane in 3 dimensions will be the solutions of one linear equation. In two dimensions one linear equation determines a line, while in three dimensions two equations are needed to determine a line.
A linear equation of the form ax + by + cz = d can be written as the dot product vr = d where v is the vector (a, b, c) and r is (x, y, z).
Thus solutions to it all have the same value of their component in the direction of v, and are not determined in directions perpendicular to, normal to, or orthogonal to v (all these words mean the same thing). Thus v is normal to a vector pointing from one solution to another.
A line can be characterized by giving the coordinates of two points on it (in any space) and a plane by giving the coordinates of three points that don't all lie on a single line.
Further we can describe a line by giving any point on it, and a vector that points in its direction.
Likewise we can describe the points on a plane by giving a point and two linearly independent vectors starting from that point that lead to other points in the plane.
We now examine the relations between the various characterizations of lines and planes.