Who invented the Dot product and Cross Product - J.W. Gibbs 1901

 

Two main kinds of vector multiplications were defined, and they were called as follows:

• The direct, scalar, or dot product of two vectors
• The skew, vector, or cross product of two vectors

https://www.quora.com/Who-invented-the-dot-product-and-cross-product

What is dot product and cross product?

A2A* Thank you.

Quaternions were introduced by Hamilton in 1843. However, such mathematical structure did not fulfill the needs of Physics, despite of its handling of four variables.

Dot and cross products were defined by J.W.Gibbs, by 1901, by extracting parts of the quaternion product. They helped somewhat to treat more properly some of the physical variables.

By 2007, I wrote in the paper Four-vector Algebra : At the beginning of the twenty century, Physics in general, and relativity theory in particular, was lacking an appropriate mathematical formalism to represent the new physical quantities that were being discovered. But, despite the fact that all physical variables such as space-time points, velocities, potentials, currents, etc., were recognized that must be represented with four values, the quaternions were not being used to represent and manipulate them.

…the quaternions were dismissed for the difficulties and complications produced by their quaternion product. With the new product, suggested in the present paper, for fourvectors, all those difficulties disappear.

The use of four-vectors allows discerning constants, variables and relations, previously unknown to Physics, which are needed to complete and make coherent the theory.

The four-vector algebra proposed in the present paper seems to be the correct mathematical tool to study the fundamental physical variables and their describing equations.

The proposed new product for four-vectors is described in other answers I have given at Quora, for example in: What is a vector divided by a vector? or in: What is a vector space in linear algebra?.

Finally, if you are interested in an interesting application of four-vectors to electromagnetism, which is mostly reformulated, please refer to my paper: Maxwell's equations from four-vectors

https://math.stackexchange.com/questions/62318/origin-of-the-dot-and-cross-product#:~:text=In%201773%2C%20Joseph%2DLouis%20Lagrange,vector%22%20and%20%22scalar%22.

Here are a few references about the history of linear algebra:

"A Brief History of Linear Algebra and Matrix Theory"

"History of Linear Algebra"

Cross Product - Wikipedia: History

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The Wikipedia article seems to address your question most directly:

History

In 1773$1773$, Joseph-Louis Lagrange introduced the component form of both the dot and cross products in order to study the tetrahedron in three dimensions.[23]${}^{[23]}$ In 1843$1843$, William Rowan Hamilton introduced the quaternion product, and with it the terms "vector" and "scalar". Given two quaternions [0,u]$[0,\mathbf{u}]$ and [0,v]$[0,\mathbf{v}]$, where u$\mathbf{u}$ and v$\mathbf{v}$ are vectors in R3${\mathbf{R}}^{\mathbf{3}}$, their quaternion product can be summarized as [−u⋅v,u×v]$[-\mathbf{u}\cdot \mathbf{v},\mathbf{u}\times \mathbf{v}]$. James Clerk Maxwell used Hamilton's quaternion tools to develop his famous electromagnetism equations, and for this and other reasons quaternions for a time were an essential part of physics education.

In 1878$1878$ William Kingdon Clifford published his Elements of Dynamic which was an advanced text for its time. He defined the product of two vectors[24]${}^{[24]}$ to have magnitude equal to the area of the parallelogram of which they are two sides, and direction perpendicular to their plane.

Oliver Heaviside and Josiah Willard Gibbs also felt that quaternion methods were too cumbersome, often requiring the scalar or vector part of a result to be extracted. Thus, about forty years after the quaternion product, the dot product and cross product were introduced — to heated opposition. Pivotal to (eventual) acceptance was the efficiency of the new approach, allowing Heaviside to reduce the equations of electromagnetism from Maxwell's original 20$20$ to the four commonly seen today.[25]${}^{[25]}$

Largely independent of this development, and largely unappreciated at the time, Hermann Grassmann created a geometric algebra not tied to dimension two or three, with the exterior product playing a central role. In 1853$1853$ Augustin-Louis Cauchy, a contemporary of Grassmann, published a paper on algebraic keys which were used to solve equations and had the same multiplication properties as the cross product.[26]${}^{[26]}$[27]${}^{[27]}$ Clifford combined the algebras of Hamilton and Grassmann to produce Clifford algebra, where in the case of three-dimensional vectors the bivector produced from two vectors dualizes to a vector, thus reproducing the cross product.

The cross notation and the name "cross product" began with Gibbs. Originally they appeared in privately published notes for his students in 1881$1881$ as Elements of Vector Analysis. The utility for mechanics was noted by Aleksandr Kotelnikov. Gibbs's notation and the name "cross product" later reached a wide audience through Vector Analysis, a textbook by Edwin Bidwell Wilson, a former student. Wilson rearranged material from Gibbs's lectures, together with material from publications by Heaviside, Föpps, and Hamilton. He divided vector analysis into three parts:

First, that which concerns addition and the scalar and vector products of vectors. Second, that which concerns the differential and integral calculus in its relations to scalar and vector functions. Third, that which contains the theory of the linear vector function.

Two main kinds of vector multiplications were defined, and they were called as follows:

• The direct, scalar, or dot product of two vectors
• The skew, vector, or cross product of two vectors

Several kinds of triple products and products of more than three vectors were also examined. The above-mentioned triple product expansion was also included.