Hi Martin !
I see i understand it coorect. Than the
"perpendicular component" + "paralel component" =
"perpendicular component" = A - "paralel component"
Little more effitient than 2 cross products.
This both is a very usefull in practice.
The "paralel component" ( projection of B into A) is usefull
in dynamics and collision solve and e.t.
The "perpendicular component" is used for distance calculation
from line to point and many many others.
Sometimes it is included as "vector" class members.
I think it is usefull post it on own web page.
For example i for a long time calculated projections using normalisation
of vector ( the unefficient with sqrt , and i did it many in dynamic calculations
), and it was a shame for me to discover the way like you post.
Where I can, I have put links to Amazon for books that are relevant to
the subject, click on the appropriate country flag to get more details
of the book or to buy it from them.
New Foundations for Classical Mechanics (Fundamental Theories of Physics). This
is very good on the geometric interpretation of this algebra. It has lots of insights
into the mechanics of solid bodies. I still cant work out if the position, velocity,
etc. of solid bodies can be represented by a 3D multivector or if 4 or 5D multivectors
are required to represent translation and rotation.