This section is divided into a number of subsections, links to which are:

Tensor products

Cross products

Triple products

Wedge products

Rotors

 

Before we dance with vector products, we remind the basic product that is not related to any vector structure. For any two sets A and B, their Cartesian product consists of all ordered pairs (𝑎, b) such that 𝑎 ∈ A and b ∈ B,
\[ A \times B = \left\{ (a,b)\,:\ a \in A, \quad b\in B \right\} . \]
If sets A and B carry some algebraic structure, as in our case, they are vector spaces, then we can define a suitable structure on the product set as well. So, direct product is like Cartesian product, but with some additional structure. In our case, we equip it with addition operation
\[ \left( a_1 , b_1 \right) + \left( a_2 , b_2 \right) = \left( a_1 + a_2 , b_1 + b_2 \right) \]
and scalar multiplication
\[ k \left( a , b \right) = \left( k\,a , k\,b \right) , \qquad k \in \mathbb{F}. \]
Here 𝔽 is a field of scalars (either ℚ, rational numbers, or ℝ, real numbers, or ℂ, complex numbers). It is a custom to denote the direct product of two or more scalar fields as 𝔽² or 𝔽n.
In 1844, Hermann Grassmann (1809--1877) published (from his own pocket) a book on geometric algebra not tied to dimension two or three. Grassmann develops several products, including a cross product represented then by brackets.

In 1877, to emphasize the fact that the result of a dot product is a scalar while the result of a cross product is a vector, William Kingdon Clifford (1845--1879) coined the alternative names scalar product and vector product for the two operations. These alternative names are still widely used in the literature.

In 1881, Josiah Willard Gibbs (1839--1903), and independently Oliver Heaviside (1850--1925), introduced the notation for both the dot product and the cross product using a period (a ⋅ b) and an "×" (a × b), respectively, to denote them.

The Italian mathematician Gregorio Ricci-Curbastro (1853--1925) and his student Tullio Levi-Civita (1873--1941) are credited for invention and popularization of tensor calculus. One of Albert "Einstein's most-notable contributions to the world of mathematics is his application of tensors in general relativity theory. Abstract mathematical formulation was done in the middle of twentieth century by Alexander Grothendieck (1928--2014).

The wedge product symbol took several years to mature from Hermann Grassmann's work (The Theory of Linear Extension, a New Branch of Mathematics, 1844) and Élie Cartan’s book on differential forms published in 1945. The wedge symbol ∧ seems to have originated with Claude Chevalley (1909--1984) sometime between 1951 and 1954 and gained widespread use after that.

Vector Products

For high-dimensional mathematics and physics, it is important to have the right tools and symbols with which to work. This section provides an introduction for constructing a large variety of vector spaces from known spaces. Besides direct products, we consider other versions of its generalizations.

Let V be a vector space over the field 𝔽, where 𝔽 is either ℚ (rational numbers) or ℝ (real numbers) or ℂ (complex numbers). The bilinear functions from V × V into 𝔽 were considered in sections regarding dot product and inner product. In special section, we consider two important vector products, known as tensor product and cross product, as well as its generalization wedge product (also known as exterior product). Our exposition is an attempt to bridge the gap between the elementary and advanced understandings of tensor product, including wedge product.

Dot Product

By the scalar product (synonymous, the dot of two vectors v = (v₁, v₂, … , vn) and u = u₁, u₂, … , un) from ℝn, denoted by v • u, we mean the scalar quantity
\[ {\bf v} \bullet {bf u} = {\bf u} \bullet {bf v} = v_1 u_1 + v_2 u_2 + \cdots + v_n u_n . \]
The dot product is a topic of special section in Part 5.

 

Tensor product

 

Vector or Cross product

 

Triple product

 

Wedge or Exterior product

 

Rotors

 

  1. With v = [ 1, 0, −1 ], find v⊗v.
  2. Find [ 1, 0, −1 ] ⊗ [ 1, 1, 1 ].

 

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