A line of infinite length is like a single dimension within a higher dimensional space. There are two cases to consider:
- The line goes through the origin the origin. In this case the line could be represented by a normalised vector with appropriate dimension for the space we are working in. If we are working in two dimensional space then we could represent the line by y=m*x. If we are working in 'n' dimensional space then we can represent the line with n-1 degrees of freedom.
- If the line does not go through the origin, then we could represent the line in the same way as above but with a translation, so it would require two vectors to define the line. If we are working in two dimensions then we could represent the line by y=a + m*x. If we are working in more than two dimensions then we could use plucker coordinates to represent the line.
A line may be defined as the shortest distance between two points (assuming that space itself is not curved). We could define such a line using 2 vectors (for instance, the two endpoints or one endpoint and the distance and direction to the other).
Euclid defined a line as "A line is breadthless length". In two dimensions a line could represent the boundary between two adjoining areas.