Some preliminaries

OK, I hear you…

But if we’re going to discuss engineering and mathematics at any level other than the most superficial then by golly we had best make sure we’re talking the same language – and in the modern world, the language of engineering and science is… math. So, if anyone told you there would be no “math” here then, well, prepare to be gravely disappointed.

Now, my hope is to keep this light and relatively short, as I fully intend to develop further mathematical preliminaries as needed. And yes, I promise, I will absolutely do my level-best to try to keep my mathematical symbology as consistent as possible as the articles are published. But one thing that needs to be stated up front is that, given the breadth and depth of the subjects we’re going to cover here, it is absolutely impossible to be entirely consistent with symbolic representation of quantities. Again, I’ll do my best, recognizing that I’m condensing dozens of lectures taught over the course of a decade or more at different times and different places.

Let’s start by talking about vectors. A point is a location in (generally three-dimensional, or 3D) space without any sense of direction. It is given in 2-dimensions (2D) by coordinates P=(a, b) and in 3-dimensions by coordinates P=(a, b, c), where we’re using “dimensions” to describe the space we’re operating in (simply, 1D is a line, 2D is a (generally planer) surface, 3D is a box).

With me so far? Good.

A vector is a mathematical device used to describe something measures the distance and direction separation two points in (generally 3D) space. As such, a vector has a magnitude and a direction. The magnitude is nothing more or less than the size of the vector, while the direction gives the orientation between the two points represented by the extent of the vector. If we denote the vector by x then the vector can be represented mathematically by…

x = (P₁, P₂) = ( (a₁, b₁, c₁) , (a₂, b₂, c₂) )

Limiting ourselves to 2D for the time being (extension to 3D is, actually, trust me, trivial), we can define a vector in terms of base (or unit) vectors that correspond to the particular coordinate system we’re employing. Further limiting ourselves for now with the Cartesian or Rectangular Coordinate System, we define these basis vectors (which have a magnitude of precisely 1.0) as i and j, representing a unit of length in transit along (respectively) the x- and y-axes of our coordinate system. Using this notation, we have a definition of a vector d in 2D given by…

d = d_x i + d_y j

The length, or magnitude, of the vector d is given by

|d| = sqrt ( d_x² + d_y² )

The addition of vectors is a way to graphically represent the something moves from point a to point b, and then moves to point c sort of problem that people always loved in grammar school during elementary algebra. The addition of vectors occurs along the components of the vectors, as such…

f = f_x i + f_y j = d + e = ( d_x + e_x) i + ( d_y + e_y) j

As such, it’s obvious that

f_x = d_x + e_x, and
f_y = d_y + e_y

Motion

With the basics of position, direction, and vectors behind us, let us turn to a brief discussion of motion. Consider a particle at a position in a 1D space (a line, in this case aligned with the x-axis in a Cartesian space) given by x₀. If the particle moves to a position given by x, it’s displacement is simply the difference between it’s current location and it’s previous, to wit (in differential form)…

dx = x – x₀

Expanding this into vector form, for a space broader than 1D, is simply a matter of writing this expression in vector form…

dx = x – x₀

… where vector subtraction works precisely the same as vector addition, only with a change in sign. If we wish to know the rate at which that displacement occurs, then we find we’re interested in the average velocity of this particle, which is defined as the displacement divided by the elapsed time…

v = dx / dt

Note that we’ve written this in differential form rather than the somewhat more traditional incremental form. As such, what we’ve actually defined is the instantaneous velocity.

Welcome to analytical calculus!

Similarly, the change in the velocity over time is the acceleration, which is defined as

a = dv / dt

I’m thinking this is as good a place as any to stop for a little break. Hopefully this discussion hasn’t been too slow for anyone. Please drop me a line and let me know either way. I’ll be back in a bit with a discussion about mass, forces, and energies.