$$
\newcommand{\RR}{\mathbb{R}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\EE}{\mathbb{E}}
\newcommand{\HH}{\mathbb{H}}
\newcommand{\SO}{\operatorname{SO}}
\newcommand{\dist}{\operatorname{dist}}
\newcommand{\length}{\operatorname{length}}
\newcommand{\uppersum}[1]{{\textstyle\sum^+_{#1}}}
\newcommand{\lowersum}[1]{{\textstyle\sum^-_{#1}}}
\newcommand{\upperint}[1]{{\textstyle\smallint^+_{#1}}}
\newcommand{\lowerint}[1]{{\textstyle\smallint^-_{#1}}}
\newcommand{\rsum}[1]{{\textstyle\sum_{#1}}}
\newcommand{\partitions}[1]{\mathcal{P}_{#1}}
\newcommand{\erf}{\operatorname{erf}}
\newcommand{\ihat}{\hat{\imath}}
\newcommand{\jhat}{\hat{\jmath}}
\newcommand{\khat}{\hat{k}}
\newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}}
\newcommand{\smat}[1]{\left(\begin{smallmatrix}#1\end{smallmatrix}\right)}
$$
In this part we acquaint ourselves with the mathematics of \(n\)-dimensional spaces. Such spaces are described using \(n\)-tuples of real numbers \[(x_1,x_2,\ldots, x_n)\] and are indespensible when discussing positions in real, physical space (especially in dimension 3 and 4 for space and spacetime). But they are also indespensible to mathematicians working on other problems, where high dimensional spaces are used to track the behavior of complex objects.
We learn to differentiate between points (which measure position) and vectors (which give a direction and magnitude), and cover the operations of dot and cross product which are essential to the geometry of vectors.
Finally, we use our knowledge of points and vectors to construct formulas for shapes in 2,3 and higher dimensional spaces; from lines planes and spheres to more complicated objects.