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A two-vector or bivector [1] is a tensor of type () and it is the dual of a two-form, meaning that it is a linear functional which maps two-forms to the real numbers (or more generally, to scalars). The tensor product of a pair of vectors is a two-vector. Then, any two-form can be expressed as a linear combination of tensor products of pairs of ...
Minkowski sums act linearly on the perimeter of two-dimensional convex bodies: the perimeter of the sum equals the sum of perimeters. Additionally, if K {\textstyle K} is (the interior of) a curve of constant width , then the Minkowski sum of K {\textstyle K} and of its 180° rotation is a disk.
In geometric algebra, a multivector is defined to be the sum of different-grade k-blades, such as the summation of a scalar, a vector, and a 2-vector. [17] A sum of only k-grade components is called a k-vector, [18] or a homogeneous multivector. [19] The highest grade element in a space is called a pseudoscalar.
Illustration of tangential and normal components of a vector to a surface. In mathematics, given a vector at a point on a curve, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the curve, called the tangential component of the vector, and another one perpendicular to the curve, called the normal component of the vector.
When working with coordinates in geometric algebra it is usual to write the basis vectors as (e 1, e 2, ...), a convention that will be used here. A vector in real two-dimensional space R 2 can be written a = a 1 e 1 + a 2 e 2, where a 1 and a 2 are real numbers, e 1 and e 2 are orthonormal basis vectors. The geometric product of two such ...
The fundamental difference is that GA provides a new product of vectors called the "geometric product". Elements of GA are graded multivectors: scalars are grade 0, usual vectors are grade 1, bivectors are grade 2 and the highest grade (3 in the 3D case) is traditionally called the pseudoscalar and designated .