Development of the idea of the Vector Cross Product

Engineering mathematics uses the vector dot product and cross product, often as mathematical definitions with useful properties. The dot product gives the vector amount that one vector contributes along the same line as another vector, and that straightforward result may serve as sufficient explanation for the dot product. The cross product, however, is partly the result of multiplying different components of two vectors to get a product vector that is at right angles to both of the original vectors and that has a magnitude equal to the area of the parallelogram that the two vectors frame. The reason for multiplying the components of the vectors in such a manner is usually not explained in terms of how the mathematics was designed to account for properties of nature. The usual explanation is that the mathematical relation will be found useful.

The dot product is

a · b = |a||b|cos(theta)

The magnitude of the vector cross product is

|a X b| = |a||b|sin(theta)

And the vector cross product is

a X b = (a2b3 - a3b2)i -(a1b3 - a3b1)j + (a1b2 - a2b1)k

The magnitude of the cross product is equivalent to the area of the parallelogram that the two vectors (a and b) describe, and the formula for the vector product is the same as the determinant for the array shown here.

    | i  j  k|
det |a1 a2 a3|
    |b1 b2 b3|

The vector product is perpendicular to both of the original vectors.

Cross Product

A few questions

How was the cross product defined?
Was it defined to account for how nature works?
Why does the cross product (and the natural field behavior) act at right angles with the plane that the original vectors define?
Is there a reason that the order of cross product coefficients is given in the determinant?

While trying to find more about the origin of the cross product in particular, I found an advertisement for a book about determinants in a 1882 book about quaternions. The advertisement promoted the book as increasing interest in the discussion through showing how natural considerations led to the development of the determinant. More generally, it seems that including the historical development or even the cause for the development of a tool shows an appropriate use of the tool or property.

The following information is from A History of Vector Analysis; The Evolution of the Idea of a Vectorial System, Michael J. Crowe, 1967, republished in 1985.

Early influences

Parallelogram of forces Bernard Lamy and Pierre Varignon
Leibniz' Concept of a Geometry of Situation
I believe that by this method one could treat mechanics almost like geometry, and one could even test the qualities of materials, because this ordinarily depends on certain figures in their sensible parts. Finally, I have no hope that we can get very far in physics until we have found some such method of abridgement to lighten its burden of imagination. (Quoted from Leibniz "Studies in Geometry of Situation with a Letter to Christian Huygens" in Philosophical Papers and Letters, 1956)
The Concept of the Geometrical Representation of Complex Numbers

Creative Design

Sir William Rowan Hamilton (1805-)

After his third year at Trinity College of Dublin University, he was "offered the honor of becoming Andrews' Professor of Astronomy at the University of Dublin and Royal Astronomer of Ireland". (MJC)
While seeking "the extension of the complex number system to three dimensions", Hamilton discovered quaternions on October 16, 1843. Quoting MJC
Thus in a very dramatic manner Hamilton discovered and announced the discovery of quaternions. These are hypercomplex numbers of the form w + ix + jy + kz, where w, x, y, and z are real numbers, and i, j, and k are unit vectors, directed along the x, y, and z axes respectively. The i, j, and k units obey the following laws: ij=k jk=i ki=j ji=-k kj=-i ik=-j ii=jj=kk=-1
It is to be noted that for two quaternions q and q', qq' does not in general equal q'q. The loss of commutativity in quaternions, while it is very important historically, is also significant mathematically, because this complicates calculations in which quaternions are used. All the other properties discussed above are satisfied by quaternions. Thus it may be verified that quaternion multiplication is associative and quaternion division is unambiguous. These are two important properties which bear special mention, since they are not preserved in the algebra of modern vectors.
32

Hermann Grassmann (1809-1877)

Attended no mathematical lectures at the University of Berlin.
Grassmann wrote Theorie der Ebbe and Flut in 1840. (The book was published in 1911).
He said that the idea of geometrical multiplication was from his father's books. His father had written, The rectangle itself is the true geometrical product, and the construction of it, as it appears in section 53, is really geometrical multiplication. If the concept of multiplication is taken in its purest and most general sense, then one comes to view a construction as something constructed from elements already constructed and in fact constructed in the same way. Thus multiplication is only a construction of a higher power. In geometry the point is the original "producing" element; from it through construction the line emerges; if we take as the basis of a new construction the finite line constructed from the point, and if we treat it in the same manner as we formerly treated the point, then the rectangle emerges. Just as the line came from the point, so the rectangle comes from the line.
Included philosophical reasoning based on space in his early work.
Quoting MJC, Grassmann's geometrical product is quite similar to the modern vector (cross) product. Another quote in the Grassmann multiplication of two vectors the product is not another vector, but rather a directed area. It is true that this area (or the set of geometrically equal areas) determines a vector (or set of vectors) perpendicular to that area, and that the vector (or vectors) so defined is precisely that vector which is the product in the modern form of the multiplication.
Grassmann defined a linear product that "is identical to the modern scalar product".

A History of Vector Analysis, James Walter Joiner, 1971, (doctoral dissertation at George Peabody College for Teachers)

The history of the development of vectors and the cross product involved more than one person and the consequent choices and practice of the community of workers who heard about and used the developing set of tools. Hamilton was one of the early developers, notably from his discovery of quaternions in 1843. His idea seems to have been based on analogies with the theory of complex or imaginary numbers (numbers having a component with a factor of the square root of -1.) Hamilton was the first to use the word vector, based on the Latin veher, "to draw".

Oliver Heaviside promoted the use of vectors (including cross products) in his 1893 book "Electromagnetic Theory". He said that vector mathematics were considerably easier to use than quaternions and that vector mathematics could be discovered while using Cartesian mathematics through noting groups of calculations that repeat when working with the math for certain kinds of engineering applications.