A while ago I asked Show whether the vector projection is idempotent, which made it clear to me that the idempotency of the vector projection could be checked with a simple substitution.
Now I am thinking about the scalar projection, which takes vectors as input and returns a scalar. This contrasts to the vector projection that took in vectors and returned a scalar. How can the scalar projection really be an idempotent map if the image is a different space than the domain?