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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.

Vector Product

For arbitrary sets A and U, we denote by V = UA the set of all functions from A into U, If U is a vector space, then the set of all U-valued functions is a vector space, independently what structure is imposed on A. Addition is the natural addition of functions in V = UA:    (f + g)(x) = f(x) + g(x), and scalar multiplication    (r f)(x) = r (f(x)), for arbitrary functions f, g and every scalar r. We leave verification that V = UA is a vector space to the reader or find a freshman whom you can charge $20 for tutoring that includes verification of all properties of vector space for V.

If A is a discrete set, then let πi be evaluation at i, so that πi(f) = f(i). Now, however, πi is vector valued rather than scalar valued, because it is a mapping from V = UA to U, and we call it the ith coordinate projection rather than the ith coordinate functional. Again these maps are all linear. In fact, the natural vector operations on UA are uniquely defined by the requirement that the projections πi all be linear. We call the value f(j) = πj(f) the jth coordinate of the vector f. Here the analogue of Cartesian n-space is the set U[1..n] of all n-tuples   α = (𝑎₁, 𝑎₂, … , 𝑎n), where [1..n] denotes the set of all integers j ∈ ℤ such that 1 ≤ j ≤ n. The set U[1..n] is also designated Un.

There is no reason why we must use the same space U at each index, as we did previously. In fact, if U₁, U₂, … , Un are any n vector spaces, then the set of all n-tuples   α = (𝑎₁, 𝑎₂, … , 𝑎n), such that 𝑎j ∈ Uj for each j is a vector space under component-wise operations of addition

\[ \alpha + \beta = \left( a_1 , a_2 , \ldots , a_n \right) + \left( b_1 , \ldots , b_n \right) = \left( a_1 + b_1 , a_2 + b_2 , \ldots , a_n + b_n \right) , \]
and scalar multiplication
\[ r\,\alpha = r \left( a_1 , a_2 , \ldots , a_n \right) = \left( r\,a_1 , r\,a_2 , \ldots , r\,a_n \right) , \qquad r \in \mathbb{F} . \]
So we obtain the Cartesian product of n vector spaces, denoted either U₁ × U₁ × × Un or ΠiUi. Since each multiple Ui is a vector space, the Cartesian product is also the vector space.    

Example 1: Let S be the unit sphere in ℝ³:

\[ S = \left\{ \mathbf{x} = \left( x_1 , x_2 , x_3 \right) \in \mathbb{R}^3 \ : \ x_1^2 + x_2^2 + x_3^2 = 1 \right\} . \]
Let Ux be the subspace of ℝ³ tangent to S at x ∈ S. This means that the vector space Ux contains the origin and by adding x it is slided to the point x in paralell way so Ux + x goes through point x.

Let

\[ U = \prod_{\mathbf{x} \in S} U_x \]
be the product of tangent planes. Then the product U consists of all functions f : S → U that assign to every point x ∈ S a vector in U, that is, a vector parallel to the tangent plane to S at x. Such a function is called a vector field on S.

The xth coordinate projection on U is evaluation at x, so πx(f) = f(x), and the natural vector operations on U are uniquely defined by the requirement that the coordinate projections all be linear. Thus, f + g must be that element of U whose value at x, πx(f + g) is πx(f) + πx(g) = f(x) + g(x) for all points on S. A similar statement is valid for scalar multiplication.    ■

End of Example 1
Theorem 1: The Cartesian product of a collection of vector spaces can be made into a vector space in exactly one way so that the coordinate projections are all linear.
With the vector operations determined uniquely as above, the proofs of all axioms for being a vector space can be verified. They did not require that the functions being added have all their values in the same space, but only that the values at a given domain element i all lie in the same space.
   

Example 2:    ■

End of Example 2

 

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

 

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