It's a more abstract kind of dimension. The vector is only one dimensional because it has a factor of one unit vector (only one associated "direction". A Bivect

It's a more abstract kind of dimension. The vector is only one dimensional because it has a factor of one unit vector (only one associated "direction". A Bivector has two factors of unit vectors to describe a 2 dimensional region. That's independent of the dimension of the space it's embedded in. So you can have a one dimensional vector in 2space, 3space, nspace etc. But it only ever describes one direction, an oriented line.