This week I am going to be doing a series on Vectors, starting off with the history of vectors.

### Complex Numbers

Vectors were born during the start of the 19th century, due to the need to represent complex numbers geometrically. The mathematicians Caspar Wessel (1745-1818), Jean Robert Argand (1768-1822) and Carl Friedrich Gauss (1777-1855) were the first to show complex numbers as being points in a two-dimensional plane, and hence as two-dimensional vectors.

This idea was part of the effort to extend two-dimensional numbers to three dimensions, however at the time no one was able to accomplish this whilst still preserving the basic algebraic properties of real and complex numbers.

In 1827, August Ferdinand Möbius published a book entitled ‘*The Barycentric Calculus’*, where he introduced line segments which had a direction and where denoted by letters. Basically, these were vectors in all but the name! In the book, he showed how to perform calculations with these line segments – how to add them and multiply them with a real number. However, these accomplishments and their importance were not noticed by the mathematical community.

### Quaternions

In 1843, William Hamilton introduced quaternions – a four dimensional system. Hamilton expressed:

“I was walking in to attend and preside along the Royal Canal, an under-current of thought was going on in my mind, which at last gave a result, whereof it is not too much to say that I felt at once the importance. An electric circuit seemed to close; and a spark flashed forth, I could not resist the impulse to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formulae.”

Quaternions were of the form: q = w + **i**x + **j**y + **k**z, where w, x, y and z are real numbers. Hamilton realised that quaternions consisted of two parts – the scalar and the vector part. Hamilton used his ‘fundamental formula’: *i*^{2} =*j*^{2} = *k*^{2} = –** ijk** = -1, to multiply them together and thus discovered that the product of quaternions was not commutative.

In the 1850s, Peter Guthrie Tait applied quaternions to problems involving electricity and magnetism and other problems in physics.

### Grassmann Algebra

Around the same time, Hermann Grassman published the book *‘The Calculus of Extension’*, which developed a new geometric calculus. He used this to simply large parts of two classical works – *Analytical Mechanics* by Lagrange and *Celestial Mechanics* by Laplace. Grassmann expanded the concept of a vector from two or three dimensions to *n* dimensions, which greatly extended the ideas of space. Furthermore, he anticipated a large amount of modern matrix, linear algebra and vector and tensor analysis. However, his work was largely ignored as it was lacking in explanatory examples and written in a strange style with overcomplicated notation.

### Gibbs

The development of the algebra of vectors and vector analysis is largely credited to J. Willard Gibbs. Although British scientists, including Maxwell, relied on Hamilton’s quaternions in order to express the dynamics of physical quantities, Gibbs was the first to note that the product of quaternions always had to be separated into two parts. Hence, calculations with quaternions introduced unnecessary complications and redundancies that could be removed. Therefore, he proposed defining distinct dot and cross products for pairs of vectors and introduced the vector notation we use today.

While working on vector analysis, Gibbs realised that his approach was similar to that of Grassmann and thus sought to publicise Grassmann’s work, stressing that is was more general than Hamilton’s quaternions.

Oliver Heaviside also developed his own vector analysis of the same style.

In the 1880s and 1890s, quaternions was eventually abandoned by physicists who preferred the vectorial approach proposed by Gibbs and Heaviside.

Hope you enjoyed today’s short introduction to vectors. Make sure you return for the rest of the series! M x

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