This is just a presentation on my history page of my webpage. To view the resources where I got my information, see my history page as well as my reference page under Final Project—Vector Mania.

Introduce my website Watch the Exert from Despicable Me on the definition of a vector I want to focus on the history portion of my website.

Have you ever wondered how imaginary numbers and complex system plane is applicable? I have and I got one answer as I was studying the history of the development of vectors. In fact, I found that the development of the complex plane and the concept of an imaginary number was the basis for the development of vectors. There are three main people responsible for the development of vectors. They are Caspar Wessel, William Rowan Hamilton, and Josiah Willard Gibbs.

Caspar Wessel: Caspar Wessel’s fame stems from a paper that’s purpose was to explain how to represent direction analytically using line segments. He demonstrated two things in this paper. How to add them (the same way we do vector addition today) and how to multiply them.

Caspar Wessel and Line Segment (basically vector) Multiplication: In his quest to develop a theory on Multiplication, Caspar Wessel developed the following coordinate system design: +1 corresponds with 0 degrees +e corresponds with 90 degrees -1 corresponds with 180 degrees -e corresponds with 270 degrees.

There is a rule that says the direction angle of the product shall equal the sum of the angles of the factors. (+1)*(+1)= +1 (+1)*(-1)= -1 (-1)*(-1)= +1 (-1)*(+e)= -e (+e)*(+e)= -1 So, Wessel proved that +e=√-1 which later became known as i. Thus the above coordinate system is the complex plane.

Wessel goes on to say in his paper that any line segments (vector) can be written in the form a+be. So to add two line segments: (a+be) + (c+de) = (a+c)+(b+d)e To multiply two line segments: (a+be)*(c+de)=(ac-bd)+(ad+bc)e Unfortunately Wessel’s ideas were forgotten /lost for a century. He had basically created the idea of vectors.

William Rowan Hamilton: Tried to extend the idea of a+bi as a vector (Note: I just replaced e with i, the accepted notation today of an imaginary number) to a complex plane of three dimensions by creating the triplet a+bi+cj where i=j=√-1. He could easily do the addition by adding like components, but he could never figure out how to multiply them. You could not just expand them because the law of moduli would not be fullfilled unless the you set the term ij=0. Hamilton did not think that was the right thing to do.

Hamilton’s Solution: Hamilton’s solution came to him as he was on a stroll with his wife along the Royal Canal in Dublin. His result invention became known as a Quaternion, or basically a vector of four dimensions. A quaternion was written q=a+bi+cj+dk where i=j=k=√- 1.

Josiah Willard Gibbs: Worked in the field of Physics Known as the founder of our modern vector system He merged together Hermann Grassman’s idea of limitless dimensionality and Hamilton’s idea of the quaternion. He dropped the constant term of the quaternion, and maintained the components with imaginary numbers, except i, j, and k became notation for x, y, and z in the ordinary Cartesian coordinate system.

Josiah Willard Gibbs: Developed cross product and dot product His system eventually won out against the quaternion because it wasn’t built in a complex plain and was thus more comprehensible. Also, his system was more flexible and dynamic in that it could be applied to more dimensions.