Vectors - Introduction

In mathematics, you can think of vectors as an ordered list. Formally, this is not really correct, but for now, this definition will do.

You will not yet be able to understand the meaning of the following sentence, but at the end of the whole course, you hopefully will

A vector is an element of a vector space.

The notation for vectors is

As shown in the example, to denote vectors, conventionally a lowercase letter with a small arrow on top of it is used. Often, instead of brackets, parentheses may be used. This is especially common in pure mathematics and is totally valid notation:

In engineering and related fields, square brackets are usually preferred. I will be using square brackets in this course, but keep in mind that parentheses are just as valid.

In the example above, we have a so-called “two-dimensional” vector $\vec{v}$, where $v_1$ is the first component of $\vec{v}$ and $v_2$ is the second component of $\vec{v}$.

Now consider a vector $\vec{v}$ defined as

Such a vector can also be represented graphically as an arrow in a coordinate system. Typically with the arrow starting at the origin $(0,0)$, and ending at the point corresponding to the vector components - in this case $(2,3)$.

The vector $\vec{v}$ above can be drawn as follows:

Similarly, a three-dimensional vector can be represented in a three-dimensional coordinate system. Four dimensional vectors can also graphically be thought of as these “arrows in some four-dimensional space”, but since we can’t draw such a space, let alone imagine it, we have to stick to 2D and 3D for the graphical representation.