Split-quaternion - Pan-orthogonality

Pan-orthogonality

When coquaternion, then the scalar part of q is w.
Definition: For non-zero coquaternions q and t we write q ⊥ t when the scalar part of the product is zero.

• For every v ∈ I, if q, t ∈ C_v, then q ⊥ t means the rays from 0 to q and t are perpendicular.
• For every p ∈ J, if q, t ∈ D_p, then q ⊥ t means these two points are hyperbolic-orthogonal.
• For every r ∈ E and every a ∈ R, p = p(a, r) and v = v(a, r) satisfy p ⊥ v.
• If u is a unit in the coquaternion ring, then q ⊥ t implies qu ⊥ tu.

Proof: follows from, which can be established using the anticommutativity property of vector cross products.