Rotors are a rotation representation generalized from Quaternions and Complex numbers that works in any numbers of dimensions.

The article starts by pointing out that rotation is defined as a scalar property on a plane. The canonical representation of a plane is a bivector representing the outer product of two vectors. The bivector is in principle a fundamental concept just like the vector. Similar to how a vector represents a point with a scalar length, a bivector represents a plane with a scalar area (signed). The area of a bivector a∧b is |a||b|sin(angle). The reason we also see the term |a||b|sin(angle) from cross product is because cross product actually gives rise to a bivector instead a vector! Historically we've been confusing bivectors with vectors because they have the same representation in 3D.

The inner and outer product completes a geometric product of two vectors: ab=a⋅b+a∧b. Reflection R(a,v) is defined elegantly using geometric product: R(a,v)=-ava. Then it's noted that two reflections is equivalent to a rotation by twice the angle between a and b: R(a,R(b,v))=ba v ab. The "ab" here is known as a rotor. Applying a rotor on both sides of a vector rotates this vector in the plane a∧b by twice the angle between a and b. Quaternion is just a representation of Rotors in 3d: i:=y∧z, j:=z∧x, k:=k∧y. The scalar part (w in w+xi+yj+zk) corresponds to the inner product.

Keywords: bivector, geometric product, rotor, quaternion, rotation

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