Skew product in coordinates for R=3?

Question

Mathematics

The coordinate method, for the dot product, can be easily derived from its properties. A coordinate skew product method, deduce as well as a scalar one, replacing the cosine property with the sine properties, for R=3 - it is impossible, only for R=2. Why can't the skew product in coordinates for R = 3 be derived in the same way as the scalar product by replacing the cosine properties with the sine properties?
PS: The skew product is the oriented area of ​​the parallelogram

Answer 1

There are two products in 2D: scalar (projection) and oblique (oriented area).
There are three products in 3D: scalar (projection), vector (oriented area) and mixed (oriented volume).
I think there are already four pieces in 4D.
About the formula. Why won't it work? It will, but you misspelled what [e ₁ e ₂ ] equals. It is equal in 3D not to one, but to e ₃ .