Since the geometric product of two even multivectors is an even multivector, they define an " even subalgebra ".
12.
The relationship between the magnitude of a multivector and the area or volume spanned by the vectors is an important feature in all dimensions.
13.
Notice that because " V " has dimension two the basis bivector is the only multivector in ? " V ".
14.
The set of multivectors on a vector space " V " is graded by the number of basis vectors that form a basis multivector.
15.
In geometric algebra, they can be further generalized to the notions of projection and rejection of a general multivector onto / from any invertible " k "-blade.
16.
:: : : Geometric algebra is one framework that systematically allows a vector to be divided by another vector, to produce a multivector that contains a scalar and a bivector component.
17.
A solution is to adopt the " overdot notation ", in which the scope of a geometric derivative with an overdot is the multivector-valued function sharing the same overdot.
18.
An example of a multivector-valued function of multivectors that is linear but is " not " an outermorphism is grade projection where the grade is nonzero, for example projection onto grade 1:
19.
There is a general method for rotating a vector involving the formation of a multivector of the form R = e ^ {-\ frac { B \ theta } { 2 } } that produces a rotation \ theta in the plane and with the orientation defined by a 2-blade B.
20.
Thus, as in the two-dimensional ( complex analysis ) case, the value of an analytic ( monogenic ) function at a point can be found by an integral over the surface surrounding the point, and this is valid not only for scalar functions but vector and general multivector functions as well.
