In this picture, so-called "vector quaternions" (that is, pure imaginary quaternions) correspond not to vectors but to bivectors – quantities with magnitudes and orientations associated with particular 2D planes rather than 1D directions. The relation to complex numbers becomes clearer, too: in 2D, with two vector directions and , there is only one bivector basis element , so only one imaginary. But in 3D, with three vector directions, there are three bivector basis elements , , , so three imaginaries.
In this picture, so-called "vector quaternions" (that is, pure imaginary quaternions) correspond not to vectors but to bivectors – quantities with magnitudes and orientations associated with particular 2D planes rather than 1D directions. The relation to complex numbers becomes clearer too: in 2D, with two vector d,irections and , there is only one bivector basis element , so only one imaginary. But in 3D, with three vector directions, there are three bivector basis elements , , , so three imaginaries.