% m211oh54.tex - The Geometry of R^n
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%\begin{titlepage}
\begin{center}\tcr{\rmH The Geometry of {\Huge$\R^n$}} \end{center} %\vspace{0pt}
\LARGE              
%\huge
  
\noindent\tcr{\huge\bf (a) Points, vectors, angles:}\vspace{-10pt}
%\exprow{1.4}
\expcol{-2pt}
$$
\begin{array}{rcll}
\tcb{\vec{x}} & \tcb{=} & \tcb{(x_1,x_2,\dots, x_n)} & \mbox{a \tcr{\em point} 
(or \tcr{\em vector}) in \mc{\R^n,}}  \\
\tcb{\vec{x}\cdot\vec{y}} & \tcb{=} & \tcb{x_1 y_1+  \dots + x_ny_n} & 
\mbox{the \tcr{\em dot product}\/ of vectors \mc{\vec{x}} and \mc{\vec{y},}} \\
\tcb{\|\vec{x}\|} &\tcb{=} & \tcb{\sqrt{\vec{x}\cdot\vec{x},}}
%\, =\, \sqrt{x_1^2+\dots+x_n^2} 
& \mbox{the \tcr{\em length} (or \tcr{\em norm}) of \mc{\vec{x},}}\\
\tcb{d(\vec{x},\vec{y})}&\tcb{=}&\tcb{\|\vec{x}-\vec{y}\|} &
\mbox{the  \tcr{\em distance}\/ between \mc{\vec{x}}  and \mc{\vec{y},}}\\
\tcb{\cos(\vec{x},\vec{y})}&\tcb{=}&\tcb{\displaystyle\frac{\vec{x}\cdot\vec{y}}
 {\|\vec{x}\|\cdot\|\vec{y}\|}}
  & \parbox{10cm}{the \tcr{\em cosine of the angle}\/ between two non-zero vectors  
\mc{\vec{x}} and \mc{\vec{y}}.}
\end{array}
$$
%\exprow{1.0}\expcol{2pt}

\noi \tcr{\bf Remark:}  To justify the name ``\mc{\cos(\vec{x}, \vec{y})}'', we need:
\begin{description}
\item[\tcr{Theorem 6}] \tcg{(Cauchy-Schwarz Inequality)} 
For all \mc{\vec{x}, \vec{y} \in \R^n,} %we have 
$$
\tcb{|\vec{x} \cdot \vec{y}| \leq \|\vec{x}\| \cdot \|\vec{y}\|.}
$$
\item[\tcr{Definition.}] The \tcr{\em angle}\/ between two non-zero vectors 
\mc{\vec{x}, \vec{y} \in \R^n} is the unique number \mc{\theta} such that
$$
   \tcb{\cos(\theta)\, = \, \cos(\vec{x}, \vec{y}),\quad \mbox{and} \quad 
0 \le \theta \le \pi.}
$$

\iffalse
\noi{\bf Remarks.} 1) When we consider angles in the plane (as in chapter 3), 
then we can also give a meaning to angles which have a value 
between $\pi$ and $2\pi$. This, however, cannot be carried over to higher dimensions 
($n > 2$).\sps

2) The names {\em sine}\/ and {\em cosine}\/ for the wave functions 
$\sin(\theta), \cos(\theta)$ come from Latin word {\em sinus} = movements of a snake.
\spm

Besides allowing us to define angles, the Cauchy-Schwarz inequality also yields 
the following result about the geometry of $\R^n$: \spm\spm
\noi \begin{minipage}{11cm}
\fi

\item[\tcr{Corollary}] \tcg{(Triangle Inequality).}
For any vectors \mc{\vec{x}, \vec{y}} and \mc{\vec{z} \in \R^n} 
%we have
$$
\tcb{d(\vec{x},\vec{y}) \leq d(\vec{x},\vec{z}) + d(\vec{y},\vec{z}).}
$$
In particular, \mc{\|\vec{x} - \vec{y}\| \le \|\vec{x}\| + \|\vec{y}\|}.

\iffalse
\end{minipage}
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\put(0,0){\makebox(0,0)[tr]{$\vec{x}$ }}
\put(3,2){\makebox(0,0)[b]{$\vec{y}$}}
\put(5,0){\makebox(0,0)[tl]{ $\vec{z}$}}
\end{picture}}
\spm
\fi

\item[\tcr{Theorem 5}] \tcg{(Law of Cosines).} 
For any \mc{\vec{x}, \vec{y} \in \R^n} we have
$$
\tcb{\|\vec{x}-\vec{y}\|^2=\|\vec{x}\|^2 
+ \|\vec{y}\|^2-2 \|\vec{x}\| \cdot \|\vec{y}\| \cos(\vec{x},\vec{y})}
$$

\item[\tcr{Corollary}] \tcg{(Rule of Pythagoras)}. 
If \mc{\vec{x} \cdot \vec{y} = 0} 
(i.e.\ if \mc{\vec{x}}  is \tcr{perpendicular} to \mc{\vec{y}}),  then 
%\begin{minipage}{13cm}
$\tcb{\|\vec{x}-\vec{y}\|^2\, =\, \|\vec{x}\|^2 + \|\vec{y}\|^2.}$

\iffalse
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\put(2,0){\makebox(0,0)[tl]{$\vec{y}$}}
\end{picture} }
\sps
\spm

\noi {\bf Remark.} Although named after Pythagoras (who lived ca.\ 600 B.C.), 
this rule was already known to the ancient Egyptians and Babylonians 
(ca.\ 2000 B.C.), and maybe even to the prehistoric Beaker People (who are 
thought to have built Stonehenge).\footnote{Cf.\ B. Van der Waerden, 
Geometry and Algebra in Ancient Civilizations, Springer Verlag, 1983.}
\spm\spm
\fi
\end{description}
\newpage

\newcommand{\hsps}{\hspace{5pt}} %small horizontal space
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\noi \tcr{\huge\bf (b) Equations of lines, planes, and linear sets:}\spm

\noi Besides points, we shall also need to consider \tcg{lines, planes} and,
 more generally, \tcg{linear sets} in \mc{\R^n}. These are defined as follows.

\begin{description}
\item[\tcr{Line:}] \tcg{through} \tcr{two} \tcg{points} 
\mc{\vec{a}, \vec{b} \in \R^n:}

\mc{L:\; \vec{x} = \vec{a} + (\vec{b} -\vec{a})t$,  \hsps $t \in \R.}
\sps

\item[\tcr{Plane:}] \tcg{through the \tcr{three} points} 
\mc{\vec{a},\vec{b},\vec{c} \in \R^n:}

\mc{S:\; \vec{x} = \vec{a} + (\vec{b}-\vec{a})t_1 
  + (\vec{c}-\vec{a})t_2, \hsps t_1,t_2 \in \R.} 

\item[\tcr{Linear set:}] \tcg{through the \mc{r+1} points} 
\mc{\vec{a}_0, \vec{a}_1,\dots,   \vec{a}_r \in \R^n:}

\mc{S:\; \vec{x} = \vec{a}_0 + (\vec{a}_1-\vec{a}_0)t_1 + \dots 
+ (\vec{a}_r-\vec{a}_0)t_r,\hsps  t_1, \dots, t_r \in \R.}\sps

\item[\tcr{Linear space}] (= a \tcg{linear set} containing the origin 
\mc{\vec{0}}): \tcg{determined (or {\em spanned}) by the \mc{r} points/vectors} 
\mc{\vec{a}_1, \dots, \vec{a}_r \in \R^n:} 

\mc{L = \langle \vec{a}_1, \dots, \vec{a}_r \rangle 
= \{\vec{x}= \vec{a}_1 t_1+ \dots + \vec{a}_r t_r:t_1, \dots, t_r 
\in \R\}.}

\tcr{\bf Note:} We also say that \mc{L} is the \tcr{\em linear span} 
of the vectors \mc{\vec{a}_1, \dots, \vec{a}_r}.

\item[\tcr{Remark.}] \tcg{Linear sets} often arise as follows. If 
%{\em \tcr{solution set}}\/ 
$$
\begin{array}{rclccclcl}
\tcb{a_{11}x_1}&\tcb{+}&\tcb{a_{12}x_2} &\tcb{+}&\tcb{\ldots} &\tcb{+}&\tcb{a_{1n}x_n}
 &\tcb{=}& \tcb{b_1} \\
\tcb{a_{21}x_1}&\tcb{+}&\tcb{a_{22}x_2} &\tcb{+}&\tcb{\ldots} &\tcb{+}&\tcb{a_{2n}x_n}
 &\tcb{=}& \tcb{b_2} \\
       &\tcb{\vdots}& &\tcb{\vdots} && \tcb{\vdots} && \tcb{\vdots} \\ 
\tcb{a_{m1}x_1}&\tcb{+}&\tcb{a_{m2}x_2} &\tcb{+}&\tcb{\ldots} &\tcb{+}&\tcb{a_{mn}x_n}
 &\tcb{=}& \tcb{b_m} 
\end{array}
$$
is a \tcr{system of \mc{m} linear equations} in \mc{n} variables (which
in \tcr{matrix form} may be written as \mc{A\vec{x} = \vec{b}}), then its 
{\em \tcr{solution set}}\/ \mc{Sol(A\vec{x} = \vec{b})} is a linear set in \mc{\R^n}. 
To transform it into a vector equation, use \tcr{row reduction} and 
\tcr{back-substitution}.
 

\end{description}

\end{document}

