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% (find-angg "LATEX/2015-2-GA-material.tex")
% (find-angg "LATEX/2015-2-GA-material.lua")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2015-2-GA-material.tex"))
% (defun d () (interactive) (find-xpdfpage "~/LATEX/2015-2-GA-material.pdf"))
% (defun e () (interactive) (find-LATEX "2015-2-GA-material.tex"))
% (defun l () (interactive) (find-LATEX "2015-2-GA-material.lua"))
% (defun u () (interactive) (find-latex-upload-links "2015-2-GA-material"))
% (find-xpdfpage "~/LATEX/2015-2-GA-material.pdf")
% (find-lualatex-links "2015-2-GA-material")
% (find-es "dednat" "GA-material-pack")
% (find-LATEX "falta-misandria-a5.tex")
% (find-LATEXfile "2014-1-GA-P2-gab.tex")
% (find-LATEXfile "2015-1-GA-P2-gabarito.tex" "\\catcode")
% (find-LATEXfile "2015-1-GA-P2-gabarito.tex" "dednat6dir =")
% (find-sh0 "cp -v ~/LATEX/2015-2-GA-material.pdf /tmp/")
% file:///home/edrx/LATEX/2015-2-GA-material.pdf
% http://angg.twu.net/LATEX/2015-2-GA-material.pdf
\documentclass[oneside]{book}
\usepackage[colorlinks]{hyperref} % (find-es "tex" "hyperref")
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{tikz}
% \usepackage{luacode}
%
\usepackage{edrx15} % (find-angg "LATEX/edrx15.sty")
\input edrxaccents.tex % (find-angg "LATEX/edrxaccents.tex")
\input edrxheadfoot.tex % (find-dn4ex "edrxheadfoot.tex")
\input istanbuldefs % (find-LATEX "istanbuldefs.tex")
%
% \input istanbuldefs.tex % (find-istfile "defs.tex")
\def\Diag#1{\directlua{tf:processuntil()}\diag{#1}}
\def\Ded #1{\directlua{tf:processuntil()}\ded{#1}}
\def\Exec#1{\directlua{tf:processuntil() #1}}
\def\Expr#1{\directlua{tf:processuntil() output(#1)}}
\def\Expr#1{\directlua{tf:processuntil() output(tostring(#1))}}
\def\expr#1{\directlua{output(tostring(#1))}}
\def\eval#1{\directlua{#1}}
\def\e{\expr}
%
\begin{document}
\catcode`\^^J=10
\directlua{dednat6dir = "dednat6/"}
\directlua{dofile(dednat6dir.."dednat6.lua")}
\directlua{texfile(tex.jobname)}
\directlua{verbose()}
%\directlua{output(preamble1)}
\def\pu{\directlua{pu()}}
{\setlength{\parindent}{0em}
\footnotesize
\par Geometria Analítica
\par PURO-UFF - 2015.2
\par Material para exercícios - Eduardo Ochs
% \par Versão: veja o pé de página % 21/dez/2015
\par Links importantes:
\par \url{http://angg.twu.net/2015.2-GA.html} (página do curso)
\par \url{http://angg.twu.net/LATEX/2015-2-GA-material.pdf}
(lista, atualizada)
\par \url{http://angg.twu.net/2015.2-GA/2015.2-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)∈\R^2}{#1}}
\def\setofet #1{\setofst{#1}{t∈\R}}
\def\setofeu #1{\setofst{#1}{u∈\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};}
% (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 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
{\setlength{\parindent}{0em}
Exercícios 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 quadrilátero...
\Reg{2c} $||\,||\uu||\,\vv|| = ||\,||\vv||\,\uu||$
\Reg{2d} Se $||\uu|| = ||\vv||$ então $(\uu-\vv)·(\uu+\vv)=0$.
\Reg{2e} $\uu·\vv=||\uu||\,||\vv||$
\Reg{2f} Se $\uu≠\vec0$ e $\uu·\vv=\uu·\ww$ então $\vv=\ww$.
\Reg{2g} $||\uu+\vv||^2 = ||\uu||^2 + 2\uu·\vv + ||\vv||^2$.
\Reg{2h} $||\uu+\vv||^2 + ||\uu+\vv||^2 = 2(||\uu||^2 + ||\vv||^2)$.
\Reg{2i} $||\uu+\vv||^2 + ||\uu-\vv||^2 = 4\uu·\vv$.
\Reg{2j} Existe uma reta que contém os pontos $A=(1,3)$, $B=(-1,2)$ e $C=(5,4)$.
\Reg{2k} O triângulo com vértices $A=(1,0)$, $B=(0,2)$ e $C=(-2,1)$ é retângulo.
\Reg{2l} Todo vetor em $\R^2$ é combinação linear de $\uu=\VEC{2,3}$, $\vv=\VEC{1,\frac32}$.
\Reg{2m} Se $\uu≠\vec0$, $\vv≠\vec0$ e $\Pr_{\vv}\uu = \vec0$ então $\uu⊥\vv$.
}
\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 gráfico 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 olhômetro o ponto $P \in r \cap s$; encontre (também no olhômetro)
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 $
$(1,4) = (3,3)+(-1)\VEC{2,-1} ∈ r$
$(1,4) = (4,1)+3\VEC{-1,1} ∈ s$
$(1,4) ∈ r∩s$
\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 olhômetro não basta, você vai precisar resolver um sistema)
\newpage
% ___
% / _ \ _ _ __ __
% | | | | | | | | \ \ / /
% | |_| | | |_| |_ \ V /
% \___( ) \__,_( ) \_/
% |/ |/
{\setlength{\parindent}{0em}
Exercício:
Em cada um 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) já está 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 notação 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 triângulos $D_iB_iC_i$ para $i=1,\ldots,7$.
\medskip
Já vimos que na passagem de um diagrama para outro as figuras - `F's e
triângulos - podem ser transladadas, ampliadas, reduzidas, amassadas,
deformadas, espelhadas...
Quais das transformações preservam distâncias ($d(P_i,Q_i) = d(P_j,Q_j)$)?
Quais das transformações preservam ângulos ($P_i\hat{Q_i}R_i = P_j\hat{Q_j}R_j$)?
}
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;
}
$
\end{document}
http://angg.twu.net/GA/lista1_GA_2011.1.pdf
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