section_path: Properties/In higher dimensions
section_name: In higher dimensions
paragraph_text: In higher dimensions
  
  In three-dimensional space, a first degree equation in the variables x, y, and z defines a plane, so two such equations, provided the planes they give rise to are not parallel, define a line which is the intersection of the planes. More generally, in n -dimensional space n −1 first-degree equations in the n coordinate variables define a line under suitable conditions.
  
  In more general Euclidean space, R (and analogously in every other affine space), the line L passing through two different points a and b is the subset L = { (1 − t) a + t b ∣ t ∈ R }. {\displaystyle L=\left\{(1-t)\,a+tb\mid t\in \mathbb {R} \right\}.} The direction of the line is from a reference point a (t = 0) to another point b (t = 1), or in other words, in the direction of the vector b − a. Different choices of a and b can yield the same line.