Your terminology is slightly off. The linear approximation to a function of 1 variable is indeed a line. The linear approximation to a function of 2 variables forms a plane. The linear approximation to a function of any number of variables is a hyperplane. Unlike the terms "line" and "plane", a hyperplane does not traditionally have a fixed dimension. Instead, the dimensionality depends on context: in , a hyperplane has one dimension; in , it has two; and, in general, a hyperplane has one fewer dimension than the ambient space (a nice term for this is "codimension "). In general, we don't bother to describe lines as "hyperplanes in space," mainly because one cannot distinguish between dimension and codimension when the ambient space has dimensions. At the same time, it's not wrong. The distinction is relevant in and above, and so we do describe planes as "hyperplanes in space."