% (find-angg "LATEX/2016-2-GA-VR.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2016-2-GA-VR.tex"))
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% http://angg.twu.net/LATEX/2016-2-GA-VR.pdf
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\def\V(#1){\VEC{#1}}
\def\pla{{\mathsf{pla}}}




%   ____      _                    _ _           
%  / ___|__ _| |__   ___  ___ __ _| | |__   ___  
% | |   / _` | '_ \ / _ \/ __/ _` | | '_ \ / _ \ 
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%                                                

{\setlength{\parindent}{0em}
\footnotesize
\par Geometria Analítica
\par PURO-UFF - 2016.2
\par VR - 23/jan/2017 - Eduardo Ochs
\par Respostas sem justificativas não serão aceitas.
\par Proibido usar quaisquer aparelhos eletrônicos.
% \par Versão: 14/mar/2016
% \par Links importantes:
% \par \url{http://angg.twu.net/2015.2-C2.html} (página do curso)
% \par \url{http://angg.twu.net/2015.2-C2/2015.2-C2.pdf} (quadros)
% \par \url{http://angg.twu.net/LATEX/2015-2-C2-material.pdf}
% \par {\tt eduardoochs@gmail.com} (meu e-mail)
}

\bsk
\bsk

\setlength{\parindent}{0em}
\def\T(Total: #1 pts){{\bf(Total: #1 pts)}}
\def\T(Total: #1 pts){{\bf(Total: #1)}}
\def\B       (#1 pts){{\bf(#1 pts)}}
% Usage:
% 1) \T(Total: 2.34 pts) Foo
% a) \B(0.45 pts) Bar

\unitlength=5pt
\def\closeddot{\circle*{0.6}}
\def\pictline#1{{\linethickness{1.0pt}\expr{Line.new(#1):pict()}}}



1) \T(Total: 2.0 pts) Verdadeiro ou falso? Justifique.

a) \B(1.0 pts) $\Pr_{\uu} (3\uu) = 3\uu$.

b) \B(1.0 pts) Se $\uu⊥\vv$ então
   $\Pr_{\uu} (4\uu+5\vv) +
    \Pr_{\vv} (4\uu+5\vv)
            = (4\uu+5\vv)$.

\bsk
\bsk

2) \T(Total: 2.0 pts) Sejam $r=\setofxyst{y=-2-2x}$, $A=(-1,5)$,
$B=(-1,4)$, $C=(-1,3)$, $D=(-1,2)$, $E=(-1,1)$, $F=(-1,0)$,
$G=(-1,-1)$, $H=(-1,-2)$, e sejam $A'$ o ponto de $r$ mais próximo de
$A$, $B'$ o ponto de $r$ mais próximo de $B$, e assim por diante.

a) \B(0.5 pts) Encontre $A'$.

b) \B(0.7 pts) Encontre $B'$.

c) \B(0.8 pts) Calcule $d(A,r)$, $d(B,r)$, $d(C,r)$, $d(D,r)$,
$d(E,r)$, $d(F,r)$, $d(G,r)$, $d(H,r)$.



\bsk
\bsk

3) \T(Total: 2.0 pts) Represente graficamente as cônicas com as
equações abaixo. Algumas são degeneradas.

\begin{tabular}[t]{lcl}
a) \B(0.2 pts) $(x-3)^2=1$ \\
b) \B(0.2 pts) $(x+y)^2=1$ \\
c) \B(0.3 pts) $(x-3)^2 + (x+y)^2 = 0$ \\
d) \B(1.0 pts) $(x-3)^2 + (x+y)^2 = 1$ \\
\end{tabular}
\quad
\begin{tabular}[t]{lcl}
e) \B(0.3 pts) $(x-3)(x+y)=0$ \\
f) \B(1.0 pts) $(x-3)(x+y) = 1$ \\
\end{tabular}




\bsk
\bsk

4) \T(Total: 2.0 pts) Uma aplicação do `$×$' que não foi mencionada na
p.35 é a seguinte. Se $A,B,C∈\R^3$, definimos:
%
$$\begin{array}{c}
  \pla(A,B,C) \;:=\; \setofxyzst{w_1x+w_2y+w_3z = w_1A_1 + w_2A_2 + w_3A_3} \\[5pt]
  \text{(onde $\ww := \Vec{AB}×\Vec{AC}$)} \\
  \end{array}
$$
%
\noindent
A aplicação é: {\sl se $A,B,C$ não são colineares então $\pla(A,B,C)$
  é o plano contendo $A$, $B$, e $C$.}

\ssk

a) \B(0.2 pts) Sejam $A=(2,0,0)$, $B=(0,3,0)$, $C=(0,0,4)$,
$π=\pla(A,B,C)$. Expresse $π$ na forma $\setofxyzst{ax+by+cz=d}$. Quem
são $a$, $b$, $c$, $d$?

b) \B(0.8 pts) Encontre o valor de $z$ que faz com que o ponto
$(1,1,z)$ pertença ao $π$ do item anterior.

c) \B(0.2 pts) Teste se $(2,3,4) ∈ \pla((1,0,0),\,(2,0,0),\,(3,0,0))$.

d) \B(0.8 pts) Explique o que acontece quando $A,B,C$ são colineares.
O que é $\pla(A,B,C)$ neste caso?


\bsk
\bsk

5) \T(Total: 2.0 pts) Calcule a distância entre
$r=\setofst{(x,4-2x,0)}{x∈\R}$ e $r'=\setofst{(3-z,3-z,z)}{z∈\R}$.





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%     \pictaxes%
%     \pictline{v(-1,0), v(1,-2), -2, 2}
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% 1) \T(Total: 2.34 pts) Foo
% a) \B(0.45 pts) Bar


\end{document}

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