The "perp dot product" a^⊥·b for a and b vectors in the plane is a modification of the two-dimensional dot product in which a is replaced by the perpendicular vector rotated 90° to the left defined by Hill. It satisfies the identities a^⊥·b | = | left bracketing bar a right bracketing bar left bracketing bar b right bracketing bar sin θ (a^⊥·b)^2 + (a·b)^2 | = | ( left bracketing bar a right bracketing bar )^2 ( left bracketing bar b right bracketing bar )^2 where θ is the angle from vector a to vector b.
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