An introduction

The present paper is intended, as far as possible, to give an advanced and exact insight into the following formulas surrounding two- and three-dimensional linear vector addition, for those who are interested in the theory or those who are presently learning the math. Regardless of your mathematical level, I find importance in revisiting basics every now and again. That said, despite if you are new to vectors or returning from a higher education, I welcome you to enjoy this paper. Because I find importance in revisiting basics, the essay will begin with the most basic form of vectors, a line, and from that single line shall come a multitude of vector equations. You may be asking yourself, where did this formula come from, what drove the author to create formulas for vectors? I had just finished a nine-hour shift at work and had begun the academic portion of my day, math class. At the time of writing, I am an attending student at Seneca College, North York, in the MIT program.

Though I am set to be a machinist of some sort, the universe of physics and mathematics has long held my attention. Even as a child, the way things worked^[1] interested me and peaked my creativity in my LEGO worlds through gear ratios and building planning. Eventually play led to sport, and sport to art, most notably in my career, traditional karate; Shorin Ruy Karate Jutsu, in which I have earned my Shodan. Karate has been single handedly the largest impact on my career to date. By achieving my Shodan I have not just ascertained a higher level of understanding of karate, but also my studying methods. This understanding of my own mastery in karate is the path I take in my attempt to master other subjects, including asking questions.

Due to a higher level of question asking, karate led back to science. I re-began my journey with the reading of Brief Answers to the Big Questions (Hawking et al., 2018), hoping to find exactly that, however I found something much more pleasing, a thirst for more. I then read A Brief History of Time (Hawking, 1998), followed by Relativity (Einstein & Lawson, 1961). But I was still parched. I read everything from Astrophysics to Quantum Mechanics in a hunt to quench my thirst. I read big names and small, but I still hungered. When I arrived at third semester mathematics in college, my focus was taken from physics to vectors. As basic as they were, I formed an understanding with them, and from this understanding came a vast enjoyment of the subject. I could picture the graphs in my mind, and class had become a movie right before my eyes. The calculations were entertaining as it was simply trigonometry, this proved to be a wonderful way to spend a Friday evening. But then it happened, my professor, Srishti, introduced a table of values to organize our calculations.

In case it has not been made clear enough, I simply do not enjoy writing a table of values when a simple equation can be used instead. I thought back to my high school physics classes thinking there would be a formula. I reached into my bag to pull out the very formula sheet that I used in high school and I looked under the collision section of the sheet, but there was nothing. There was no equation for vector calculations. Disappointed, I completed the class by reluctantly drawing the table of values for each question. At the end I asked Srishti if I could use the whiteboard, and in that moment, I began the construction of two equations to solve the resultant and angle of one or more two dimensional vectors. In short, I succeeded. From my success came the drive to create three more equations to solve the resultant and angles of one or more three dimensional vectors. The task was completed over the following reading week.

October 2023

Calvin Hendrik Thomas Smit