Warning: this is an htmlized version!
The original is across this link,
and the conversion rules are here.
% (find-angg "LATEX/2016-1-GA-material.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2016-1-GA-material.tex"))
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% (defun e () (interactive) (find-LATEX "2016-1-GA-material.tex"))
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% (find-sh0 "cp -v  ~/LATEX/2016-1-GA-material.pdf /tmp/pen/")
%   file:///home/edrx/LATEX/2016-1-GA-material.pdf
%               file:///tmp/2016-1-GA-material.pdf
%           file:///tmp/pen/2016-1-GA-material.pdf
% http://angg.twu.net/LATEX/2016-1-GA-material.pdf
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\usepackage{amssymb}
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%
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\input edrxaccents.tex            % (find-angg "LATEX/edrxaccents.tex")
\input edrxchars.tex              % (find-LATEX "edrxchars.tex")
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\catcode`\^^J=10
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%\def\eval#1{\directlua{#1}}
%\def\pu{\directlua{pu()}}

\directlua{dofile "edrxtikz.lua"} % (find-LATEX "edrxtikz.lua")
% (find-dn6 "picture.lua" "V")
%L V.__tostring = function (v) return format("(%.3f,%.3f)", v[1], v[2]) end
%L V.__div      = function (v, k) return v*(1/k) end
%L V.__index.tow = function (A, B, t) return A+(B-A)*t   end  -- towards
%L V.__index.mid = function (A, B)    return A+(B-A)*0.5 end  -- midpoint
%L V.__index.norm = function (v) return math.sqrt(v[1]*v[1] + v[2]*v[2]) end
%L V.__index.rotleft = function (vv) return v(-vv[2], vv[1]) end
%L 
\def\e{\expr}





%   ____      _                    _ _           
%  / ___|__ _| |__   ___  ___ __ _| | |__   ___  
% | |   / _` | '_ \ / _ \/ __/ _` | | '_ \ / _ \ 
% | |__| (_| | |_) |  __/ (_| (_| | | | | | (_) |
%  \____\__,_|_.__/ \___|\___\__,_|_|_| |_|\___/ 
%                                                

{\setlength{\parindent}{0em}
\footnotesize
\par Geometria AnalÃtica
\par PURO-UFF - 2016.1
\par Material para exercÃcios - Eduardo Ochs
% \par VersÃo: veja o pà de pÃgina % 21/dez/2015
\par Links importantes:
\par \url{http://angg.twu.net/2016.1-GA.html} (pÃgina do curso)
\par \url{http://angg.twu.net/LATEX/2016-1-GA-material.pdf}
     (lista, atualizada)
\par \url{http://angg.twu.net/2016.1-GA/2016.1-GA.pdf} (quadros)
\par \url{http://angg.twu.net/2015.1-GA/GA_Reis_Silva.pdf} (livro)
\par \url{http://angg.twu.net/2015.1-GA/mariana_imbelloni_retas.pdf}
\par {\tt eduardoochs@gmail.com} (meu e-mail)
}

\bsk
\bsk

%      _       __     
%   __| | ___ / _|___ 
%  / _` |/ _ \ |_/ __|
% | (_| |  __/  _\__ \
%  \__,_|\___|_| |___/
%                     

% Dots, labels, vectors
%
\def\uu{\vec u}
\def\vv{\vec v}
\def\ww{\vec w}
\def\VEC#1{{\overrightarrow{(#1)}}}

\def\nm#1{\|#1\|}
\def\Reg#1{(#1)}

\def\setofxyst#1{\setofst{(x,y)â\R^2}{#1}}
\def\setofet  #1{\setofst{#1}{tâ\R}}
\def\setofeu  #1{\setofst{#1}{uâ\R}}
\def\setofpt  #1 #2 #3 #4 {\setofet{(#1,#2)+t\VEC{#3,#4}}}
\def\setofpu  #1 #2 #3 #4 {\setofeu{(#1,#2)+u\VEC{#3,#4}}}

%  _   _ _         
% | |_(_) | __ ____
% | __| | |/ /|_  /
% | |_| |   <  / / 
%  \__|_|_|\_\/___|
%                  

% \mygrid and \myaxes
% (find-es "tikz" "mygrid")
\tikzset{mycurve/.style=very thick}
\tikzset{axis/.style=semithick}
\tikzset{tick/.style=semithick}
\tikzset{grid/.style=gray!20,very thin}
\tikzset{anydot/.style={circle,inner sep=0pt,minimum size=1.2mm}}
\tikzset{opdot/.style={anydot, draw=black,fill=white}}
\tikzset{cldot/.style={anydot, draw=black,fill=black}}
%
\def\mygrid(#1,#2) (#3,#4){
  \clip              (#1-0.4, #2-0.4) rectangle (#3+0.4, #4+0.4);
  \draw[step=1,grid] (#1-0.2, #2-0.2) grid      (#3+0.2, #4+0.2);
  \draw[axis] (-10,0) -- (10,0);
  \draw[axis] (0,-10) -- (0,10);
  \foreach \x in {-10,...,10} \draw[tick] (\x,-0.2) -- (\x,0.2);
  \foreach \y in {-10,...,10} \draw[tick] (-0.2,\y) -- (0.2,\y);
}
\def\myaxes(#1,#2) (#3,#4){
  \clip              (#1-0.4, #2-0.4) rectangle (#3+0.4, #4+0.4);
 %\draw[step=1,grid] (#1-0.2, #2-0.2) grid      (#3+0.2, #4+0.2);
  \draw[axis] (-20,0) -- (20,0);
  \draw[axis] (0,-20) -- (0,20);
  \foreach \x in {-20,...,20} \draw[tick] (\x,-0.2) -- (\x,0.2);
  \foreach \y in {-20,...,20} \draw[tick] (-0.2,\y) -- (0.2,\y);
}

% Grid color
\tikzset{grid/.style=gray!50,very thin}

\def\tikzp#1{\mat{\begin{tikzpicture}#1\end{tikzpicture}}}

\def\mydraw       #1;{\draw [mycurve]  \expr{#1};}
\def\mydot        #1;{\node [cldot] at \expr{#1} [] {};}
\def\myldot #1 #2 #3;{\node [cldot] at \expr{#1} [label=#2:${#3}$] {};}
\def\myseg     #1 #2;{\draw [mycurve]  \expr{#1} -- \expr{#2};}
\def\mylabel #1 #2 #3;{\node []     at \expr{#1} [label=#2:${#3}$] {};}
\def\myseggrid  #1 #2;{\draw [grid]    \expr{#1} -- \expr{#2};}





%  ____            _             _     _       
% |  _ \ ___  __ _(_)_ __   __ _| | __| | ___  
% | |_) / _ \/ _` | | '_ \ / _` | |/ _` |/ _ \ 
% |  _ <  __/ (_| | | | | | (_| | | (_| | (_) |
% |_| \_\___|\__, |_|_| |_|\__,_|_|\__,_|\___/ 
%            |___/                             

{\setlength{\parindent}{0em}

ExercÃcios de V/F/justifique da primeira lista do Reginaldo, reescritos:

\Reg{2a} Se $α\uu+β\vv=\vec0$ entÃo $α=0$ e $β=0$.

\Reg{2b} Seja $ABCD$ um quadrilÃtero...

\Reg{2c} $||\,||\uu||\,\vv|| = ||\,||\vv||\,\uu||$

\Reg{2d} Se $||\uu|| = ||\vv||$ entÃo $(\uu-\vv)Â(\uu+\vv)=0$.

\Reg{2e} $\uuÂ\vv=||\uu||\,||\vv||$

\Reg{2f} Se $\uuâ\vec0$ e $\uuÂ\vv=\uuÂ\ww$ entÃo $\vv=\ww$.

\Reg{2g} $||\uu+\vv||^2 = ||\uu||^2 + 2\uuÂ\vv + ||\vv||^2$.

\Reg{2h} $||\uu+\vv||^2 + ||\uu+\vv||^2 = 2(||\uu||^2 + ||\vv||^2)$.

\Reg{2i} $||\uu+\vv||^2 + ||\uu-\vv||^2 = 4\uuÂ\vv$.

\Reg{2j} Existe uma reta que contÃm os pontos $A=(1,3)$, $B=(-1,2)$ e $C=(5,4)$. 

\Reg{2k} O triÃngulo com vÃrtices $A=(1,0)$, $B=(0,2)$ e $C=(-2,1)$ Ã retÃngulo. 

\Reg{2l} Todo vetor em $\R^2$ Ã combinaÃÃo linear de $\uu=\VEC{2,3}$, $\vv=\VEC{1,\frac32}$. 

\Reg{2m} Se $\uuâ\vec0$, $\vvâ\vec0$ e $\Pr_{\vv}\uu = \vec0$ entÃo $\uuâ\vv$.
 
}


\newpage

%  ____            _                          
% |  _ \ _ __ ___ (_) ___  ___ ___   ___  ___ 
% | |_) | '__/ _ \| |/ _ \/ __/ _ \ / _ \/ __|
% |  __/| | | (_) | |  __/ (_| (_) |  __/\__ \
% |_|   |_|  \___// |\___|\___\___/ \___||___/
%               |__/                          

A {\sl projeÃÃo sobre $\vv$ de $\ww$}, $\Pr_{\vv} \ww$, Ã sempre um
vetor da forma $»\vv$.

Digamos que $\Pr_{\vv} \ww_1 = »_1 \vv_1$, $\Pr_{\vv} \ww_2 = »_1 \vv_2$, etc.

Determine $»_1$, $»_2$, etc.

%L p = function (a, b) return O + a*uu + b*vv end
%L O, uu, vv = v(3, 1), v(2, 1), v(-1, 1)

% mypgrid
%
\def\mypgrid#1;{
  \myseggrid p(-3,-#1) p(-3,#1);
  \myseggrid p(-2,-#1) p(-2,#1);
  \myseggrid p(-1,-#1) p(-1,#1);
  \myseggrid p(0,-#1) p(0,#1);
  \myseggrid p(1,-#1) p(1,#1);
  \myseggrid p(2,-#1) p(2,#1);
  \myseggrid p(3,-#1) p(3,#1);
  %
  \myseggrid p(-#1,-3) p(#1,-3);
  \myseggrid p(-#1,-2) p(#1,-2);
  \myseggrid p(-#1,-1) p(#1,-1);
  \myseggrid p(-#1,0) p(#1,0);
  \myseggrid p(-#1,1) p(#1,1);
  \myseggrid p(-#1,2) p(#1,2);
  \myseggrid p(-#1,3) p(#1,3);
}
\def\drawvec #1 #2;{
  \draw [->] \e{#1} -- \e{#2};
}
\def\drawlvec #1 #2 #3 #4;{
  \draw [->] \e{#1} -- \e{#2};
  \mylabel {#2} {#3} {#4};
}

a)
%L O, uu, vv = v(1, 1), v(2, 0), v(0, 2)
\pu
$\tikzp{[scale=0.35,auto]
    \myaxes (-8,-8) (10,10);
    \mypgrid 3;
    \drawvec p(1,1) p(2,2);
    \drawlvec p(0,0) p(2,0) 0 \vv;
    \drawlvec p(0,0) p(3,1) 0 \ww_1;
    \drawlvec p(0,0) p(3,2) 0 \ww_2;
    \drawlvec p(0,0) p(3,3) 45 \ww_3;
    \drawlvec p(0,0) p(2,3) 90 \ww_4;
    \drawlvec p(0,0) p(1,3) 90 \ww_5;
    \drawlvec p(0,0) p(0,3) 90 \ww_6;
    \drawlvec p(0,0) p(-1,3) 90 \ww_7;
    \drawlvec p(0,0) p(-2,3) 90 \ww_8;
    \drawlvec p(0,0) p(-3,3) 135 \ww_9;
    \drawlvec p(0,0) p(-3,2) 180 \ww_{10};
    \drawlvec p(0,0) p(-3,1) 180 \ww_{11};
    \drawlvec p(0,0) p(-3,0) 180 \ww_{12};
    \drawlvec p(0,0) p(-3,-1) 180 \ww_{13};
    \drawlvec p(0,0) p(-3,-2) 180 \ww_{14};
    \drawlvec p(0,0) p(-3,-3) 225 \ww_{15};
    \drawlvec p(0,0) p(-2,-3) 270 \ww_{16};
    \drawlvec p(0,0) p(-1,-3) 270 \ww_{17};
    \drawlvec p(0,0) p(0,-3)  270 \ww_{18};
    %
  }
$

b)
%L O, uu, vv = v(1, 1), v(1, 1), v(-1, 1)
\pu
$\tikzp{[scale=0.45,auto]
    \myaxes (-8,-8) (10,10);
    \mypgrid 3;
    \drawvec p(1,1) p(2,2);
    \drawlvec p(0,0) p(2,0) 45 \vv;
    \drawlvec p(0,0) p(3,1) 45 \ww_1;
    \drawlvec p(0,0) p(3,2) 45 \ww_2;
    \drawlvec p(0,0) p(3,3) 90 \ww_3;
    \drawlvec p(0,0) p(2,3) 135 \ww_4;
    \drawlvec p(0,0) p(1,3) 135 \ww_5;
    \drawlvec p(0,0) p(0,3) 135 \ww_6;
    \drawlvec p(0,0) p(-1,3) 135 \ww_7;
    \drawlvec p(0,0) p(-2,3) 135 \ww_8;
    \drawlvec p(0,0) p(-3,3) 180 \ww_9;
    \drawlvec p(0,0) p(-3,2) 225 \ww_{10};
    \drawlvec p(0,0) p(-3,1) 225 \ww_{11};
    \drawlvec p(0,0) p(-3,0) 225 \ww_{12};
    \drawlvec p(0,0) p(-3,-1) 225 \ww_{13};
    \drawlvec p(0,0) p(-3,-2) 225 \ww_{14};
    \drawlvec p(0,0) p(-3,-3) 270 \ww_{15};
    \drawlvec p(0,0) p(-2,-3) 315 \ww_{16};
    \drawlvec p(0,0) p(-1,-3) 315 \ww_{17};
    \drawlvec p(0,0) p(0,-3)  315 \ww_{18};
    %
  }
$




% \end{document}

\newpage


Calcule:

$\{x:\{0,1,2,3\}; x^2\}$

$\{x:\{0,1,2,3\}, xâ2; x\}$

\msk

Represente graficamente:

$A := \{(1,4), (2,4), (1,3)\}$

$B := \{(1,3), (1,4), (2,4)\}$

$C := \{(1,3), (1,4), (2,4), (2,4)\}$

$D := \{(1,3), (1,4), (2,3), (2,4)\}$

$h := \{(0,3), (1,2), (2,1), (3,0)\}$

$k := \{x:\{0,1,2,3\}; (x,3-x)\}$

$m := \{y:\{0,1,2,3\}; (3-y, y)\}$


% (Adaptado do material da optativa de lÃgica que eu tou dando...)

\newpage

Let

$A = \{x:\{-1,...,4\}; x^2\}$ and

$B = \{x:\{-1,...,4\}; x^2â5; x\}$.

Then $A$ and $B$ can be calculated by:

\msk

$\begin{array}{cc}
 x & x^2 \\ \hline
 -1 & 1 \\
  0 & 0 \\
  1 & 1 \\
  2 & 4 \\
  3 & 9 \\
  4 & 16 \\
 \end{array}
 \qquad
 \begin{array}{cccc}
 x & x^2 & x^2â5 & x \\ \hline
 -1 &  1 & 1 & -1 \\
  0 &  0 & 1 & 0 \\
  1 &  1 & 1 & 1 \\
  2 &  4 & 1 & 2 \\
  3 &  9 & 0 & \\
  4 & 16 & 0 & \\
  \end{array}
$

\msk

We get:

$A = \{1,0,1,4,9,16\}$,

$B = \{-1,0,1,2\}$.


\bsk

Let

$A = \{x:\{1,...,5\}, y:\{1,...,x\}, x+yâ6; (x,y)\}$ and

$B = \{y:\{1,...,5\}, x:\{y,...,5\}, x+yâ6; (x,y)\}$.

Then $A$ and $B$ can be calculated by:

\msk

$\begin{array}{ccccc}
 x & y & x+y & x+yâ6 & (x,y) \\ \hline
 1 & 1 &  2  &   1   & (1,1) \\
 2 & 1 &  3  &   1   & (2,1) \\
   & 2 &  4  &   1   & (2,2) \\
 3 & 1 &  4  &   1   & (3,1) \\
   & 2 &  5  &   1   & (3,2) \\
   & 3 &  6  &   1   & (3,3) \\
 4 & 1 &  5  &   1   & (4,1) \\
   & 2 &  6  &   1   & (4,2) \\
   & 3 &  7  &   0   &       \\
   & 4 &  8  &   0   &       \\
 5 & 1 &  6  &   1   & (5,1) \\
   & 2 &  7  &   1   &       \\
   & 3 &  8  &   0   &       \\
   & 4 &  9  &   0   &       \\
   & 5 & 10  &   0   &       \\
 \end{array}
 \qquad
 \begin{array}{ccccc}
 y & x & x+y & x+yâ6 & (x,y) \\ \hline
 1 & 1 &  2  &   1   & (1,1) \\
   & 2 &  3  &   1   & (2,1) \\
   & 3 &  4  &   1   & (3,1) \\
   & 4 &  5  &   1   & (4,1) \\
   & 5 &  6  &   1   & (5,1) \\
 2 & 2 &  4  &   1   & (2,2) \\
   & 3 &  5  &   1   & (3,2) \\
   & 4 &  6  &   1   & (4,2) \\
   & 5 &  7  &   0   &       \\
 3 & 3 &  6  &   1   & (3,3) \\
   & 4 &  7  &   0   &       \\
   & 5 &  8  &   0   &       \\
 4 & 4 &  8  &   0   &       \\
   & 5 &  9  &   0   &       \\
 5 & 5 & 10  &   0   &       \\
 \end{array}
$

\msk

We get:

$A = \{ (1,1), (2,1), (2,2), (3,1), (3,2), (3,3), (4,1), (4,2), (5,1)\}$ and

$B = \{ (1,1), (2,1), (3,1), (4,1), (5,1), (2,2), (3,2), (4,2), (3,3)\}$.

\bsk





\newpage

%  ____      _            
% |  _ \ ___| |_ __ _ ___ 
% | |_) / _ \ __/ _` / __|
% |  _ <  __/ || (_| \__ \
% |_| \_\___|\__\__,_|___/
%                         

% (find-fline       "~/2015.2-GA/")
% (find-djvupage    "~/2015.2-GA/2015.2-GA.djvu")

{\bf 2)} (Fizemos este em sala em 16/dez/2015)

Represente graficamente as retas abaixo.

Nas parametrizadas indique no grÃfico os pontos associados a $t=0$ e $t=1$.

$r_a = \setofxyst{ x+2y=0 }$

$r_b = \setofxyst{ x+2y=4 }$

$r_c = \setofxyst{ x+2y=2 }$

$r_d = \setofxyst{ 2x+3y=0 }$

$r_e = \setofxyst{ 2x+3y=6 }$

$r_f = \setofxyst{ 2x+3y=3 }$

$r_g = \setofpt 3 -1 -1 1 $

$r_h = \setofpt 3 -1 -2 1 $

$r_i = \setofpt 3 -1 1 -1 $

$r_j = \setofpt 0 3 2 0 $

$r_k = \setofpt 2 0 0 1 $

$r_l = \setofxyst{ y=4 }$

$r_m = \setofxyst{ y=4+x }$

$r_n = \setofxyst{ y=4-2x }$


\bsk
\bsk

%  ____                                _        _              _           
% |  _ \ __ _ _ __ __ _ _ __ ___   ___| |_ _ __(_)______ _  __| | __ _ ___ 
% | |_) / _` | '__/ _` | '_ ` _ \ / _ \ __| '__| |_  / _` |/ _` |/ _` / __|
% |  __/ (_| | | | (_| | | | | | |  __/ |_| |  | |/ / (_| | (_| | (_| \__ \
% |_|   \__,_|_|  \__,_|_| |_| |_|\___|\__|_|  |_/___\__,_|\__,_|\__,_|___/
%                                                                          

%L r0, rv = v(2,3), v(1,1)
%L s0, sw = v(2,3), v(2,-1)
%L rt = function (t) return r0 + t*rv end
%L su = function (u) return s0 + u*sw end
\pu
\def\rt#1{\expr{rt(#1):xy()}}
\def\su#1{\expr{su(#1):xy()}}

% \rt 0 \rt 1 \rt 2
% \su 0 \su 1 \su 2

{\bf 3)}
Em cada um dos casos abaixo, represente $r$ e $s$ graficamente,
marcando os pontos associados a $t=0$, $t=1$, $u=0$, $u=1$; encontre
no olhÃ∧metro o ponto $P \in r \cap s$; encontre (tambÃm no olhÃ∧metro)
os valores de $t$ e $u$ associados a $P$; e verifique que você
encontrou o $t$ e o $u$ certos, fazendo como abaixo.

\msk

%L inter  = v(1,4)
%L r0, rv = v(3,3), v(2,-1)
%L s0, sw = v(4,1), v(-1,1)
\pu
% (find-pgfmanualpage  44 "3.9    Adding Labels Next to Nodes")
% (find-pgfmanualtext  44 "3.9    Adding Labels Next to Nodes")
$\tikzp{[scale=0.5,auto]
    \mygrid (-1,-1) (7,5);
    \draw[mycurve] \rt{-2} -- \rt{5};
    \draw[mycurve] \su{-2} -- \su{5};
    \node [cldot] at \rt{0} [label=60:${t{=}0}$] {};
    \node [cldot] at \rt{1} [label=60:${t{=}1}$] {};
    \node [cldot] at \su{0} [label=200:${u{=}0}$] {};
    \node [cldot] at \su{1} [label=200:${u{=}1}$] {};
    \node [cldot] at \su{3} [label=60:$P$] {};
  }
$

$r = \setofpt 3 3 2 -1 $

$s = \setofpu 4 1 -1 1 $

$(1,4) = (3,3)+(-1)\VEC{2,-1} â r$

$(1,4) = (4,1)+3\VEC{-1,1} â s$

$(1,4) â râs$

\msk

a) $r = \setofpt 1 0 0 3 $, $s = \setofpu 0 4 2 0 $

b) $r = \setofpt 1 0 3 1 $, $s = \setofpu 0 2 2 3 $

c) $r = \setofet{ (1+3t,t) }$, $s = \setofeu{ (2u,2+3u) } $

d) $r = \setofpt 0 3 2 -1 $, $s = \setofpu 1 0 1 3 $

(No d o olhÃ∧metro nÃo basta, você vai precisar resolver um sistema)

% \end{document}

\newpage


%   ___                     
%  / _ \    _   _    __   __
% | | | |  | | | |   \ \ / /
% | |_| |  | |_| |_   \ V / 
%  \___( )  \__,_( )   \_/  
%      |/        |/         

{\setlength{\parindent}{0em}

ExercÃcio:

Em cada uma das figuras abaixo vamos definir o sistema de coordenadas
$Σ$ por

$Σ=(O,\uu,\vv)$ e

$(a,b)_Σ = O+a\uu+b\vv$.

Sejam:

$B = (1,3)_Σ$, $C = (3,3)_Σ$,

$D = (1,2)_Σ$, $E = (2,2)_Σ$,

$A = (1,1)_Σ$.

Desenhe a figura formada pelos pontos $A$, $B$, $C$, $D$ e $E$ e pelos
segmentos de reta $\overline{AB}$, $\overline{BC}$ e $\overline{DE}$.

(O item (a) jà està feito.)

}

% myvgrid
%
\def\myvgrid{
  \myseggrid p(0,0) p(0,4);
  \myseggrid p(1,0) p(1,4);
  \myseggrid p(2,0) p(2,4);
  \myseggrid p(3,0) p(3,4);
  \myseggrid p(4,0) p(4,4);
  \myseggrid p(0,0) p(4,0);
  \myseggrid p(0,1) p(4,1);
  \myseggrid p(0,2) p(4,2);
  \myseggrid p(0,3) p(4,3);
  \myseggrid p(0,4) p(4,4);
  \draw [->] \expr{p(0,0)} -- \expr{p(0,1)};
  \draw [->] \expr{p(0,0)} -- \expr{p(1,0)};
}
\def\mytriangle{
    \myseg p(1,2) p(1,3);
    \myseg p(1,3) p(3,3);
    \myseg p(3,3) p(1,2);
    \mydot p(1,2);
    \mydot p(1,3);
    \mydot p(3,3);
}


%L p = function (a, b) return O + a*uu + b*vv end

a)
%L O, uu, vv = v(3, 1), v(2, 1), v(-1, 1)
\pu
$\tikzp{[scale=0.4,auto]
    \myaxes (-1,-1) (13,9);
    \myvgrid
    \mylabel p(0,0) 270 O;
    \mylabel p(1,0)   0 \uu;
    \mylabel p(0,1) 180 \vv;
    %
    \myseg p(1,1) p(1,3);
    \myseg p(1,3) p(3,3);
    \myseg p(1,2) p(2,2);
    \myldot p(1,3) 180 B; \myldot p(3,3) 0 C;
    \myldot p(1,2) 180 D; \myldot p(2,2) 0 E;
    \myldot p(1,1) 180 A;
  }
$
%
\quad
%
b)
%L O, uu, vv  = v(2, 2), v(1, 0), v(0, 1)
\pu
$\tikzp{[scale=0.4,auto]
    \myvgrid; \myaxes (-1,-1) (6,6);
    \mylabel p(0,0) 270 O;
    \mylabel p(1,0)   0 \uu;
    \mylabel p(0,1)  90 \vv;
  }
$

c)
%L O, uu, vv  = v(-5, 1), v(2, 0), v(0, 1)
$\tikzp{[scale=0.3,auto] \pu
    \myvgrid; \myaxes (-6,-1) (4,6);
    \mylabel p(0,0) 270 O;
    \mylabel p(1,0)   0 \uu;
    \mylabel p(0,1)  90 \vv;
  }
$
%
\quad
%
d)
%L O, uu, vv = v(1, 1), v(1, 0), v(0, 2)
$\tikzp{[scale=0.3,auto] \pu
    \myvgrid; \myaxes (-1,-1) (6,10);
    \mylabel p(0,0) 270 O;
    \mylabel p(1,0)   0 \uu;
    \mylabel p(0,1)  90 \vv;
  }
$
%
\quad
%
e)
%L O, uu, vv = v(2, 2), v(0, 1), v(1, 0)
$\tikzp{[scale=0.4,auto] \pu
    \myvgrid; \myaxes (-1,-1) (6,6);
    \mylabel p(0,0) 270 O;
    \mylabel p(1,0)  90 \uu;
    \mylabel p(0,1)   0 \vv;
  }
$


f)
%L O, uu, vv = v(4, 4), v(-2, 1), v(-1, -2)
$\tikzp{[scale=0.3,auto] \pu
    \myvgrid; \myaxes (-8,-5) (6,8);
    \mylabel p(0,0)   0 O;
    \mylabel p(1,0) 180 \uu;
    \mylabel p(0,1)   0 \vv;
  }
$
%
\quad
%
g)
%L O, uu, vv = v(-3, 1), v(1, 0), v(1, 1)
$\tikzp{[scale=0.4,auto] \pu
    \myvgrid; \myaxes (-3,-1) (6,6);
    \mylabel p(0,0) 270 O;
    \mylabel p(1,0)   0 \uu;
    \mylabel p(0,1)  90 \vv;
  }
$



\newpage



%   ___                         _        _                   _           
%  / _ \    _   _    __   ___  | |_ _ __(_) __ _ _ __   __ _| | ___  ___ 
% | | | |  | | | |   \ \ / (_) | __| '__| |/ _` | '_ \ / _` | |/ _ \/ __|
% | |_| |  | |_| |_   \ V / _  | |_| |  | | (_| | | | | (_| | |  __/\__ \
%  \___( )  \__,_( )   \_/ (_)  \__|_|  |_|\__,_|_| |_|\__, |_|\___||___/
%      |/        |/                                    |___/             

{\setlength{\parindent}{0em}

Agora vamos usar uma notaÃÃo um pouco mais pesada...

$Σ_i=(O_i,\uu_i,\vv_i)$,

$Σ_0=((0,0),\VEC{1,0},\VEC{0,1})$,

$(a,b)_{Σ_i} = O_i+a\uu_i+b\vv_i$,

$B_i = (1,3)_{Σ_i}$, $C_i = (3,3)_{Σ_i}$,

$D_i = (1,2)_{Σ_i}$, $E_i = (2,2)_{Σ_i}$,

$A_i = (1,1)_{Σ_i}$.

As figuras abaixo representam os triÃngulos $D_iB_iC_i$ para $i=1,\ldots,7$.

\medskip

JÃ vimos que na passagem de um diagrama para outro as figuras - `F's e
triÃngulos - podem ser transladadas, ampliadas, reduzidas, amassadas,
deformadas, espelhadas...

Quais das transformaÃÃes preservam distÃncias ($d(P_i,Q_i) = d(P_j,Q_j)$)?

Quais das transformaÃÃes preservam Ãngulos ($P_i\hat{Q_i}R_i = P_j\hat{Q_j}R_j$)?

}

a)
%L O, uu, vv = v(3, 1), v(2, 1), v(-1, 1)
\pu
$\tikzp{[scale=0.4,auto]
    \myaxes (-1,-1) (13,9);
    \myvgrid
    \mylabel p(0,0) 270 O_1;
    \mylabel p(1,0)   0 \uu_1;
    \mylabel p(0,1) 180 \vv_1;
    %
    \mytriangle;
    % \myseg p(1,1) p(1,3);
    % \myseg p(1,3) p(3,3);
    % \myseg p(1,2) p(2,2);
    % \myldot p(1,3) 180 B; \myldot p(3,3) 0 C;
    % \myldot p(1,2) 180 D; \myldot p(2,2) 0 E;
    % \myldot p(1,1) 180 A;
  }
$
%
\quad
%
b)
%L O, uu, vv  = v(2, 2), v(1, 0), v(0, 1)
\pu
$\tikzp{[scale=0.4,auto]
    \myvgrid; \myaxes (-1,-1) (6,6);
    \mylabel p(0,0) 270 O_2;
    \mylabel p(1,0)   0 \uu_2;
    \mylabel p(0,1)  90 \vv_2;
    \mytriangle;
  }
$

c)
%L O, uu, vv  = v(-5, 1), v(2, 0), v(0, 1)
$\tikzp{[scale=0.3,auto] \pu
    \myvgrid; \myaxes (-6,-1) (4,6);
    \mylabel p(0,0) 270 O_3;
    \mylabel p(1,0)   0 \uu_3;
    \mylabel p(0,1)  90 \vv_3;
    \mytriangle;
  }
$
%
\quad
%
d)
%L O, uu, vv = v(1, 1), v(1, 0), v(0, 2)
$\tikzp{[scale=0.3,auto] \pu
    \myvgrid; \myaxes (-1,-1) (6,10);
    \mylabel p(0,0) 270 O_4;
    \mylabel p(1,0)   0 \uu_4;
    \mylabel p(0,1)  90 \vv_4;
    \mytriangle;
  }
$
%
\quad
%
e)
%L O, uu, vv = v(2, 2), v(0, 1), v(1, 0)
$\tikzp{[scale=0.4,auto] \pu
    \myvgrid; \myaxes (-1,-1) (6,6);
    \mylabel p(0,0) 270 O_5;
    \mylabel p(1,0)  90 \uu_5;
    \mylabel p(0,1)   0 \vv_5;
    \mytriangle;
  }
$


f)
%L O, uu, vv = v(4, 4), v(-2, 1), v(-1, -2)
$\tikzp{[scale=0.3,auto] \pu
    \myvgrid; \myaxes (-8,-5) (6,8);
    \mylabel p(0,0)   0 O_6;
    \mylabel p(1,0) 180 \uu_6;
    \mylabel p(0,1)   0 \vv_6;
    \mytriangle;
  }
$
%
\quad
%
g)
%L O, uu, vv = v(-3, 1), v(1, 0), v(1, 1)
$\tikzp{[scale=0.4,auto] \pu
    \myvgrid; \myaxes (-4,-1) (6,6);
    \mylabel p(0,0) 270 O_7;
    \mylabel p(1,0)   0 \uu_7;
    \mylabel p(0,1)  90 \vv_7;
    \mytriangle;
  }
$





%  _   _ _                 _           _           
% | | | (_)_ __   ___ _ __| |__   ___ | | ___  ___ 
% | |_| | | '_ \ / _ \ '__| '_ \ / _ \| |/ _ \/ __|
% |  _  | | |_) |  __/ |  | |_) | (_) | |  __/\__ \
% |_| |_|_| .__/ \___|_|  |_.__/ \___/|_|\___||___/
%         |_|                                      

\newpage

\def\xx{\vec{x}}
\def\yy{\vec{y}}

% (find-LATEX "edrxtikz.lua" "Hyperbole.fromOxe")
%L H = Hyperbole.fromOxe(v(0,0), v(1,0), 2, 4)
%L H = Hyperbole.fromOxe(v(0,0), v(1,0), 3, 6)
%L PP(H)
\pu
$\tikzp{[scale=0.5,auto]
    \myaxes (-5,-9) (5,9);
    \myldot H.F1 135 F_1;    \myldot H.F2  45 F_2;
    \myldot H.P1 135 P_1;    \myldot H.P2  45 P_2;
    \myldot H.P3 225 P_3;    \myldot H.P4 315 P_4;
    \myldot H.P5 135 P_5;    \myldot H.P6  45 P_6;
    \myldot H.D1 225 {};     \myldot H.D2 315 {};
    \myldot H.D0 315 {};
    \mydraw H:draw();
    \mydraw H.au:draw();
    \mydraw H.av:draw();
    \mydraw H.d1:draw();
    \mydraw H.d2:draw();
  }
$



$\def\so{{\sqrt{8}}}
 \def\f{\frac}
 %
 \begin{array}{lllll}
   e = 3                       &                              &&                      &                      \\
 \xx = \VEC{1,0}               &                              &&                      &                      \\
 \yy = \VEC{0,1}               &                              && \yy = \xx'           &                      \\
 a   = 1/2                     &                              && a   = ||\xx||/2      &                      \\
 b   = \sqrt{8} / 2            &                              && b   = \sqrt{e^2-1}Âa &                      \\
 c   = 3/2                     &                              && c   = e  a          &                      \\
 \uu = \VEC{1/2,-\so/2}        &                              && \uu = a\xx - b\yy    &                      \\
 \vv = \VEC{1/2, \so/2}        &                              && \vv = a\xx + b\yy    &                      \\
 P_1 = (-1,0)                  & P_2 = (1,0)                  && P_1 = O-\xx          & P_2 = O+\xx          \\
 F_1 = (-3,0)                  & F_2 = (3,0)                  && F_1 = O-e\xx         & F_2 = O+e\xx         \\
 D_1 = (-\f 1 3, 0)            & D_2 = (\f 1 3, 0)            && D_1 = O-\f 1 e \xx   & D_2 = O+\f 1 e \xx   \\
 P_3 = (-3, 8)                 & P_4 = (3, 8)                 && P_3 = F_1+(e^2-1)\yy & P_4 = F_2+(e^2-1)\yy \\
 P_5 = (-3, -8)                & P_6 = (3, -8)                && P_5 = F_1-(e^2-1)\yy & P_6 = F_2-(e^2-1)\yy \\
 d_1 : (-\f 1 3, y)            & d_2 : (\f 1 3, y)            && d_1 : D_1+t\yy       & d_2 : D_2+t\yy       \\
 \aa_{\uu}:(\f12 t, -\f\so2 t) & \aa_{\vv}:(\f12 t, \f\so2 t) && \aa_{\uu}:O+t\uu     & \aa_{\vv}:O+t\vv     \\
 D_0 = O                       &                              && D_0 = O              &                      \\
 d_0 : D_0 + t\yy              &                              && d_0 : D_0 + t\yy     &                      \\
 \end{array}
$


% $H = \setofxyst{}$




\newpage

\def\mc#1{\multicolumn{2}{c}{#1}}
\def\f{\frac}


Elipses:

Nomes para os pontos mais interessantes:

$\begin{array}[t]{ccccccc}
     &     &     & P_3 \\
 D_1 & P_1 & F_1 &  O  & F_2 & P_2 & D_2 \\
     &     &     & P_4 \\
 \end{array}
$

\bsk

FÃrmulas para os pontos quando $P_1=(-1,0)$ e $P_2=(1,0)$:

$\begin{array}[t]{ccccccc}
     &     &     & (0,b) \\
 (-\frac1c,0) & (-1,0) & (-c,0) & (0,0)  & (c,0) & (1,0) & (\frac1c,0) \\
     &     &     & (0,-b) \\
 \end{array}
$

onde $b^2 + c^2 = a^2 = 1$.

\bsk



Uma elipse com $e=3$, $d(P,F_1)+d(P,F_2)=2$, $d(P,d_1)=3d(P,F_1)$:

$\begin{array}[t]{ccccccc}
        &        &          & (0,\f{â8}3) \\
 (-3,\_) & (-1,0) & (-\f13,0) & (0,0) & (\f13,0) & (1,0) & (3,\_) \\
        &        &          & 0,-\f{â8}3) \\
 \end{array}
$

\bsk

Uma elipse com $e=3$, $d(P,F_1)+d(P,F_2)=2$, $d(P,d_1)=\f23 d(P,F_1)$:

$\begin{array}[t]{ccccccc}
        &        &          & (0,\f{â5}3) \\
 (-\f32,\_) & (-1,0) & (-\f23,0) & (0,0) & (\f23,0) & (1,0) & (\f32,\_) \\
        &        &          & 0,-\f{â5}3) \\
 \end{array}
$

\bsk

Uma elipse com $e=3$, $d(P,F_1)+d(P,F_2)=2$, $d(P,d_1)=\f{100}{99} d(P,F_1)$:

$\begin{array}[t]{ccccccc}
        &        &          & (0,\f{â{199}}{100}) \\
 (-\f{100}{99},\_) & (-1,0) & (-\f{99}{100},0) & (0,0) & (\f{99}{100},0) & (1,0) & (\f{100}{99},\_) \\
        &        &          & 0,\f{â{199}}{100}) \\
 \end{array}
$


\newpage

HipÃrboles:


\bsk

Nomes para os pontos mais interessantes:

$\begin{array}[t]{ccccccc}
  O-»\uu & &                  &       & &   & O+»\vv \\
   P_4 & &                  &       & &   & P_5 \\
           & \mc{O-\uu}   &       & \mc{O+\vv}          \\
    F_1 & P_1 & D_1 & O & D_2 & P_2 & F_2 \\
           & \mc{O-\vv}   &       & \mc{O+\uu}          \\
   P_6 & &                  &       & &   & P_7 \\
 O-»\vv & &                  &       & &   & O+»\vv \\
 \end{array}
$

\bsk

Uma com $e=3$, $d(P,F_2)=3d(P,d_2)$, $d(P,F_2)-d(P,F_1) = \pm 2$:

$\begin{array}[t]{ccccccc}
 (-3,3â8) & &                  &       & &   & (3,3â8) \\
   (-3,8) & &                  &       & &   & (3,8) \\
           & \mc{(-1/2,â8/2)}   &       & \mc{(1/2,â8/2)}          \\
    (-3,0) & (-1,0) & (-1/3,\_) & (0,0) & (1/3,\_) & (1,0) & (3,0) \\
           & \mc{(-1/2,-â8/2)}   &       & \mc{(1/2,-â8/2)}          \\
   (-3,-8) & &                  &       & &   & (3,-8) \\
 (-3,-3â8) & &                  &       & &   & (3,-3â8) \\
 \end{array}
$



\end{document}



%  _   _ _                 _           _           
% | | | (_)_ __   ___ _ __| |__   ___ | | ___  ___ 
% | |_| | | '_ \ / _ \ '__| '_ \ / _ \| |/ _ \/ __|
% |  _  | | |_) |  __/ |  | |_) | (_) | |  __/\__ \
% |_| |_|_| .__/ \___|_|  |_.__/ \___/|_|\___||___/
%         |_|                                      

%L e = math.sqrt(5)
%L e = 2.2
%L e = 2.1
%L F1 = v(-e*e, 0)
%L P2 = v(-e, 0)
%L D2 = v(-1, 0)
%L D  = v(1, 0)
%L P3 = v(e, 0)
%L F2 = v(e*e, 0)
%L h  = 1
%L H = Hyperbole.new(v(0,0), v(e/2, h), v(e/2,-h), 2)
\pu
$\tikzp{[scale=1.2,auto]
    \myaxes (-5,-2) (5,2);
    \myldot F1  45 F_1;    \myldot F2 135 F_2=F;
    \myldot F1 315 -e^2;   \myldot F2 225 e^2;
    \myldot P2  45 P_2;    \myldot P3 135 P_3;
    \myldot P2 315 -e;     \myldot P3 225 e; 
    \myldot D2  45 D';     \myldot D  135 D;
    \myldot D2 315 -1;     \myldot D  225 1;
    \mydraw H:draw();
  }
$

\end{document}

\newpage




%  _____ _       
% | ____| |_ ___ 
% |  _| | __/ __|
% | |___| || (__ 
% |_____|\__\___|
%                

%L A, O, B, C = v(0,5), v(0,0), v(2,1), v(2,0)
%L print(A:mid(B), "hiiiiiiii")
\pu

$\tikzp{[scale=0.4,auto]
    % \myaxes (-1,-1) (13,9);
    \clip (-1,-1) rectangle (4,6);
    % \myseg A B;
    \draw [mycurve] \e{B} -- \e{C} -- \e{O} -- \e{A} -- \e{B} -- \e{O};
    % \mylabel B+(C-B)/2 0 hello;
    \mylabel A:mid(O)  180 h;
    \mylabel A:mid(C)  0 hc;
    \mylabel O:mid(B) 90 hs;
    % \myvgrid
    % \mylabel p(0,0) 270 O;
    % \mylabel p(1,0)   0 \uu;
    % \mylabel p(0,1) 180 \vv;
    %
    % \myseg p(1,1) p(1,3);
    % \myseg p(1,3) p(3,3);
    % \myseg p(1,2) p(2,2);
    % \myldot p(1,3) 180 B; \myldot p(3,3) 0 C;
    % \myldot p(1,2) 180 D; \myldot p(2,2) 0 E;
    % \myldot p(1,1) 180 A;
  }
$

\end{document}



\newpage




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