Reflection of a vector
If we want to reflect the vector with respect to the plane identified by the normal vector unit , we can break it down into the components and the reflected vector will keep the tangent component, reversing the normal one:
where the normal component to the plane is given by the dot product of over :
expressing the scalar product as a semisum of the symmetrical part of the geometric product:
Reflection of a bivector
Now suppose we reflect a bivector B, formed for example by with respect to a plane identified by the normal . We do this by reflecting and :
remembering the expression we get:
So be careful: the transformation formula is the same, but in the case of a bivector there is no minus sign. This, however, explains the difference between vectors and bivectors in the parity transformation, which consists precisely in specular reflection.
In general, for the reflection of a multivector of degree on a multivector of degree we will have:
For example, trivectors transform by reflection in the same way as vectors, i.e. they change sign (chirality).