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\begin{document}

\title{Analitik Geometri (MAT 104)\\
Final S\i nav\i\ \color{blue} \c C\"oz\"umleri
}
\date{27 May\i s 2015}
\author{David Pierce}
%\abstract{}
\maketitle
\thispagestyle{empty}

\begin{problem}
  Bir $A$ noktas\i n\i n kutupsal koordinatlar\i\ $(a,\alpha)$ olsun,
ve $B$ noktas\i n\i nki $(b,\beta)$ olsun.
\begin{enumerate}[(a)]
\item 
$(a,\alpha)=(2,\uppi/4)$ ve $(b,\beta)=(1,7\uppi/12)$ ise
$AB$ do\u grusunun uzunlu\u gu ka\c ct\i r?
\item
Genel durumda, u\c c noktalar\i n\i n koordinatlar\i na g\"ore,
$AB$ do\u grusunun uzunlu\u gu nedir?
\end{enumerate}
\end{problem}

\begin{solution}
  \begin{asparaenum}[(a)]
    \item
      \begin{minipage}{8cm}
$\angle AOB=7\uppi/12-\uppi/4=\pi/3$,
$OA=2$ ve $OB=1$ oldu\u gundan $AOB$ \"u\c cgeni ``30--60--90'' \"u\c cgendir,
dolay\i s\i yla
$AB=\surd3$.  [\.Ikinci \c s\i k da kullan\i labilir.]
      \end{minipage}\hfill
      \begin{minipage}{2cm}
\hfill
      \begin{pspicture}(-0.259,-0.5)(1.414,1.414)
        \pspolygon(0,0)(1.414,1.414)(-0.259,0.966)
\psline{->}(0,0)(0,1.414)
\psline{->}(-0.259,0)(1.414,0)
\uput[d](0,0){$O$}
\uput[r](1.414,1.414){$A$}
\uput[l](-0.259,0.966){$B$}
      \end{pspicture}
      \end{minipage}
\item
  Kosin\"us Teoremine g\"ore
  \begin{equation*}
    AB^2=a^2+b^2-2ab\cos(\beta-\alpha).
  \end{equation*}
  \end{asparaenum}
\end{solution}

\newpage
\begin{problem}
A\c sa\u g\i daki \c sekillerde, $\theta=\pm\phi$ do\u grular\i,
ba\c slang\i\c c noktas\i nda,
 ``limason'' [salyangoz] e\u grilerine te\u get ge\c cer.
Limasonlar\i n biri, 
$r=1/2+\cos\theta$ kutupsal denklemi taraf\i ndan tan\i mlan\i r.
Bu durumda $\phi$ a\c c\i s\i\ nedir?
(Hesaplamalar\i n\i z\i\ g\"osterin.)
\begin{figure}[h]
\psset{unit=1.6cm}
  \begin{pspicture}(-1.5,-1.2)(1.5,1.2)
%\psgrid
    \parametricplot0{360}{0.5 t cos add t cos mul
                          0.5 t cos add t sin mul}
\psline{->}(-1,0)(2,0)
\psline{->}(0,-1.2)(0,1.2)
\uput[dl](-0.577,1){$\theta=\phi$}
\uput[ul](-0.577,-1){$\theta=-\phi$}
\psset{linestyle=dashed}
\psline(-0.69,1.2)(0.69,-1.2)
\psline(0.69,1.2)(-0.69,-1.2)
  \end{pspicture}
\hfill%\psset{unit=1.71cm}
  \begin{pspicture}(-1.2,-1.2)(2,1.2)
%\psgrid
    \parametricplot0{360}{0.707 t cos add t cos mul
                          0.707 t cos add t sin mul}
\psline{->}(-1.2,0)(2,0)
\psline{->}(0,-1.2)(0,1.2)
\uput[ur](-1,1){$\theta=\phi$}
\uput[dr](-1,-1){$\theta=-\phi$}
\psset{linestyle=dashed}
\psline(-1.2,1.2)(1.2,-1.2)
\psline(1.2,1.2)(-1.2,-1.2)
  \end{pspicture}
  %\caption{}
  
\end{figure}
\end{problem}

\begin{solution}
  $\begin{gathered}
   1/2+\cos\phi=0,\\
\cos\phi=-1/2,\\  
\phi=\pm2\uppi/3.
  \end{gathered}$

\c Sekle g\"ore $\phi=2\uppi/3$ (veya $-\pi/3$).

  \begin{pspicture}(-4,-1.2)(1.5,1.2)
%\psgrid
    \parametricplot0{360}{0.5 t cos add t cos mul
                          0.5 t cos add t sin mul}
\psline{->}(-1,0)(2,0)
\psline{->}(0,-1.2)(0,1.2)
\uput[dl](-0.577,1){$\theta=2\uppi/3$}
%\uput[ul](-0.577,-1){$\theta=-\phi$}
\psset{linestyle=dashed}
\psline(-0.69,1.2)(0.69,-1.2)
%\psline(0.69,1.2)(-0.69,-1.2)
  \end{pspicture}
\end{solution}

\newpage

\begin{problem}
  \begin{asparaenum}[(a)]
\item
\c Sekillerde
\begin{compactitem}
  \item
Bir do\u gru, $ABC$ \"u\c cgeninin kenarlar\i n\i\
$D$, $E$, ve $F$'de kessin;
\item
$AB$, \c cemberin \c cap\i\ olsun, ve $DG\perp AB$ olsun;
\item
$DH\perp DE$ ve $DH\cdot DE=DG^2$ olsun;
\item
$EK$, $DH$'ye paralel olsun ve $FH$'yi $K$'de kessin;
\item
$CL$, $DE$'ye paralel olsun ve $AB$'yi $L$'de kessin.
\end{compactitem}
Ortadaki \c sekli tamamlay\i n.
\begin{figure}[h]
\centering
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\uput[ul](0,1){$C$}
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\uput[d](0,0){$D$}
\uput[30](0.333,0.667){$E$}
\uput[r](1,2){$F$}
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\uput[r](1.133,0.267){$K$}
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\uput[d](-0.5,0){$L$}
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\uput[r](1,0){$B$}
\uput[u](0,0){$D$}
\uput[d](0,-1){$G$}
\end{pspicture}

%\hfill
\begin{pspicture}(-2,-1.6)(1.467,1)
\begin{solution}
\psline(0,0)(-2,-1)
\uput[d](-2,-1){$F$}
\psline(-1,0)(-2,-1)
\psline(0,0)(0.6,-1.2)
\uput[d](0.6,-1.2){$H$}
\psline(-2,-1)(1.467,-1.267)
\psline(0.667,0.333)(1.467,-1.267)
\uput[d](1.467,-1.267){$K$}
\psline(-1,0)(-2,0)
\psline(0,1)(-2,0)
\uput[d](-2,0){$L$}
\end{solution}
\pspolygon(-1,0)(1,0)(0,1)
\uput[ul](-1,0){$A$}
\uput[r](1,0){$B$}
\uput[30](0,1){$C$}
\psline(0,0)(0.667,0.333)
\uput[u](0,0){$D$}
\uput[ur](0.667,0.333){$E$}
\end{pspicture}

\begin{solution}
  [Bu \c s\i k, telafi s\i nav\i ndayd\i.]
\end{solution}
\end{figure}
\item
\c Simdi $a$, $b$, $c$, ve $d$,
herhangi uzunluklar olsun.
Tan\i m\i m\i za g\"ore ne zaman $a:b::c:d$?  
Do\u gru \c sekil \c cizmek yeterlidir.
  \end{asparaenum}
\end{problem}

\begin{solution}
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  \begin{pspicture}(0,-1)(5,3)
    \pspolygon(0,0)(4,0)(4,3)
\psline(3.5,0)(3.5,0.5)(4,0.5)
\uput[d](2,0){$b$}
\uput[r](4,1.5){$a$}
\psarc(0,0){1}0{36.86}
%\uput{12pt}[18.4](0,0){$\theta$}
  \end{pspicture}
\psset{unit=5mm}
  \begin{pspicture}(0,-0.8)(4.8,3)
    \pspolygon(0,0)(4,0)(4,3)
\psline(3.6,0)(3.6,0.4)(4,0.4)
\uput[d](2,0){$d$}
\uput[r](4,1.5){$c$}
\psarc(0,0){0.8}0{36.86}
%\uput{12pt}[18.4](0,0){$\theta$}
  \end{pspicture}\hfill
[Bu \c s\i k, ikinci s\i navdayd\i.]
\end{solution}

\newpage

\begin{problem}
Bu problemde $xyz$ eksenleri birbirine diktir,
ve birim uzunlu\u gu se\c cilmi\c stir.
  \begin{enumerate}[(a)]
  \item 
$(1,2,3)$, $(1,1,1)$, ve $(-1,2,-3)$ noktalar\i ndan ge\c cen d\"uzleme
dik olan (ve s\i f\i r olmayan) bir vekt\"or bulun.
\item
$\displaystyle\frac{x-10}7=\frac{y+100}{-1}=\frac{z+1000}3$ 
do\u grusuna dik olan,
$(2,5,4)$ noktas\i ndan ge\c cen d\"uzlemin denklemini
$ax+by+cz=d$ bi\c ciminde yaz\i n.
  \end{enumerate}
\end{problem}
    
\begin{solution}
  \begin{asparaenum}[(a)]
    \item
$\begin{aligned}[t]
&\phantom{{}={}}\bigl((1,2,3)-(1,1,1)\bigr)\times\bigl((-1,2,-3)-(1,1,1)\bigr)\\
&=(0,1,2)\times(-2,1,-4)\\
&=\left(
      \begin{vmatrix}
        1&2\\1&-4
      \end{vmatrix},
      -\begin{vmatrix}
       0&2\\-2&-4 
      \end{vmatrix},
      \begin{vmatrix}
        0&1\\-2&1
      \end{vmatrix}\right)\\
&=(-4-2,-(-(-4)),-(-2))\\
&=(-6,-4,2).
    \end{aligned}$

[Cevap, $(6,4,-2)$, $(3,2,-1)$, ve saire olabilir.]
\item
  $\begin{aligned}[t]
    7x-y+3z
&=(7,-1,3)\cdot(2,5,4)\\
&=14-5+12\\
&=21.
  \end{aligned}$
  \end{asparaenum}
\end{solution}


\end{document}
