[AB] = ABsin(€) or
When two lines meet they create a form. Placing a unit circle on the centre of meeting allows me to Mark out 2 arcs of a defined length in radians. If I "produce" the lines beyond the centre, the point of meeting, to meet the circle I separate the larger arc into 3.
This construction occurs so often in geometry that until Grassmann it was taken for granted. Grassmann defined it as the inner and outer product, and the relationship is calculated, as in all geometry by trigonometry and Pythagorean theory.
The names relste to the exterior and interior angles of a closed form, but the products themselves are not tied to anything other than the meeting of two lines and their extension beyond the point of meeting.
Meeting and joining are similar activities of relation, crossing is different,but again related at the point of meeting or join.
Of course lines and points do not exist other than in my subjective experience, and even then as abstract relations or tools of measurement and comparison. They are paradigms or rules of conduct with regard to measuring and construction, and should not be confused with reality. They derive from a much broader class of actions and activities which contextualise them and give them actionable meaning. We of course have been taught to call these underpinning rules " approximations" , but as usual we promote the lesser over the greater! Approximations are that large field of constructive activity that enables engineering and any science. It is pure pragmatism and stands apart from the " ideal" , which is to say the " idea" – l.
With the outer and inner product grassmann was able to notate general rules or theorems of construction, as well as theoretical relationships in geometry In an elegant way that utilises algebra at a sophisticated level. He could now rewrite all of Euclidean geometry algebraically,tying it to the calculus of trigonometry, which is the bedrock of the calculus of differentials and integrals.
The Grassmann Algebra is really the Euclid- Grassmann algebra and as such has an inherent Pythagorean metric.