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-rw-r--r--mat/domace_naloge/28/dokument.aux44
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+\usetikzlibrary{calc} %% not really needed............. idk.
+%\usetikzlibrary{external}
+% \usetkzobj{all} % tkz-euclide > 3.02 tega ne potrebuje več (:
+%\tikzexternalize
+\sisetup{output-decimal-marker = {,}, quotient-mode=fraction,per-mode=fraction} % per-mode=symbol
+\newcommand\ddfrac[2]{\frac{\displaystyle #1}{\displaystyle #2}}
+\newcommand{\functionSamples}{100} % fix to fancier value upon release, keep low during development
+\newcommand{\razhroscevanje}{1}
+\definecolor{codegreen}{rgb}{0,0.6,0}
+\definecolor{codered}{rgb}{1,0,0}
+\definecolor{codegray}{rgb}{0.5,0.5,0.5}
+\definecolor{codepurple}{rgb}{0.58,0,0.82}
+\definecolor{backcolour}{rgb}{0.95,0.95,0.92}
+\renewcommand{\dateseparator}{. }
+\settimeformat{hhmmsstime}
+\lstdefinestyle{mystyle}{
+ backgroundcolor=\color{backcolour},
+ commentstyle=\color{codegreen},
+ keywordstyle=\color{magenta},
+ numberstyle=\tiny\color{codegray},
+ stringstyle=\color{codepurple},
+ basicstyle=\ttfamily\footnotesize,
+ breakatwhitespace=false,
+ breaklines=true,
+ captionpos=b,
+ keepspaces=true,
+ numbers=left,
+ numbersep=5pt,
+ showspaces=false,
+ showstringspaces=false,
+ showtabs=false,
+ tabsize=2
+}
+\lstset{style=mystyle}
+\def\@maketitle{%
+ \newpage
+ \null
+ \vskip 2em%
+ \begin{center}%
+ \let \footnote \thanks
+ {\LARGE \@title \par}%
+ \vskip 1.5em%
+ {\large
+ \lineskip .5em%
+ \begin{tabular}[t]{c}% <------
+ \@author% <------ Authors
+ \end{tabular}\par}% <------
+ \vskip 1em%
+ {\large \@date}%
+ \end{center}%
+ \par
+ \vskip 1.5em}
+%opening
+\newcommand{\snovdn}{Skalarni produkt vektorjev v pravokotnem koordinatnem sistemu }
+\newcommand{\predmdn}{mat}
+\newcommand{\predmkaj}{domace\_naloge}
+\newcommand{\stevilkadn}{28}
+\newcommand{\cm}[1]{\SI{#1}{\centi\meter}}
+\newcommand{\kmh}[1]{\SI{#1}{\kilo\meter\per\hour}}
+\makeatletter
+\newcommand{\xslalph}[1]{\expandafter\@xslalph\csname c@#1\endcsname}
+\newcommand{\@xslalph}[1]{%
+ \ifcase#1\or a\or b\or c\or \v{c}\or d\or e\or f\or g\or h\or i%
+ \or j\or k\or l\or m\or n\or o\or p\or r\or s\or \v{s}%
+ \or t\or u\or v\or z\or \v{z}
+ \else\@ctrerr\fi%
+}
+\AddEnumerateCounter{\xslalph}{\@xslalph}{m}
+\makeatother
+\newcommand\gauss[2]{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))} % Gauss function, parameters mu and sigma
+\newcommand*\textfrac[2]{
+ \frac{\text{#1}}{\text{#2}}
+}
+\NewTasks[counter-format=\xslalph*),label-format=\bfseries]{primerTasks}(3)
+\newcommand\vektor{\overrightarrow}
+\newcommand{\iic}{I\textsuperscript{2}C }
+\title{%
+ \snovdn --- \stevilkadn. domača naloga
+ \\
+ \large Matematika, Gimnazija Bežigrad}
+\author{\begin{tabular}{rl}
+ \textbf{Profesor:} & prof. Vilko Domajnko \\
+ \textbf{Avtor:} & Anton Luka Šijanec, 2. a
+% \textbf{Avtor:} & Anton Luka Šijanec \\ & Member 2 \\ & Member 3
+\end{tabular}}
+\newcommand\hcancel[2][black]{\setbox0=\hbox{$0#2$}%
+\rlap{\raisebox{.45\ht0}{\textcolor{#1}{\rule{\wd0}{1pt}}}}#2}
+\everymath{\displaystyle} % https://tex.stackexchange.com/a/32847/212260
+\begin{document}
+\maketitle
+\begin{abstract}
+Ta dokument vsebuje navodila in rešitve domačih nalog snovi \textit{\snovdn}pri matematiki, ki sem jih spisal sam.
+\end{abstract}
+\paragraph{Navodilo naloge} \textbf{vaje2}: 93 / 612, 622, 632, 642, 652 \textit{(+ 617, 627)}
+%\tableofcontents
+\begin{enumerate}[label=\textbf{\arabic*.}]
+ \setcounter{enumi}{611}
+ \item Izračunaj.
+ \begin{tasks}[label=\textbf{\xslalph*)}](2)
+ \task $(2,3)+(4,5)=(6,8)$ \\
+ $(2,3)-(4,5)=(-2,-2)$ \\
+ $(2,3)\cdot(4,5)=23$ \\
+ $2\cdot(4,5)=(8,10)$
+ \task $(1,2,3)+(9,8,7)=(10,10,10)$ \\
+ $(1,2,3)-(9,8,7)=(-8,-6,-4)$ \\
+ $(1,2,3)\cdot(9,8,7)=9+16+21$ \\
+ $5\cdot(9,8,7)=(45,40,35)$
+ \end{tasks}
+ \item Izračunaj.
+ \begin{tasks}[label=\textbf{\xslalph*)}](2)
+ \task $(3,7)\cdot(2,9)=69$
+ \task $(-6,10)\cdot(3,2)=2$
+ \task $(1,2,3)\cdot(4,5,6)=32$
+ \task $(-2,2,-4)\cdot(1,0,3)=-14$
+ \task $\left(5,0,1\frac{1}{2}\right)\cdot\left(1\frac{1}{2},\frac{2}{7},3\right)=12$
+ \task $\left(\sqrt{2}+\sqrt{3},2\sqrt{3}-1,-\sqrt{2}\right)\cdot\left(\sqrt{3}-\sqrt{2},\sqrt{2},\sqrt{3}\right)
+ =1-\sqrt{2}+\sqrt{6}$
+ \task $\left(2\Vec{i}+3\Vec{j}+\Vec{k}\right)\cdot\left(5\Vec{i}+\Vec{j}+3\Vec{k}\right)=16$
+ \task $\left(\Vec{i}-\Vec{j}+2\Vec{k}\right)\cdot\left(\Vec{i}+3\Vec{j}-3\Vec{k}\right)=-8$
+ \task $\left(5\Vec{i}+2\Vec{j}\right)\cdot\left(4\Vec{i}-3\Vec{j}\right)=14$
+ \task $\left(\Vec{i}-5\Vec{j}\right)\cdot\left(-\Vec{i}+3\Vec{j}\right)=-16$
+ \task $\left(2\Vec{i}-3\Vec{k}+\Vec{j}\right)\cdot\left(2\Vec{j}-3\Vec{k}\right)=11$
+ \task $\left(3\Vec{j}-2\Vec{i}-\Vec{k}\right)\cdot\left(\Vec{i}+\frac{1}{2}\Vec{j}+\frac{3}{2}\Vec{k}\right)=-2$
+ \end{tasks}
+ \setcounter{enumi}{616}
+ \item Dana sta vektorja $\Vec{m}=(3,5)$ in $\Vec{n}=(-3,2)$. Vektorja nariši v koordinatni sistem in na sekundo natančno izračunaj kot med njima.
+ \begin{multicols}{2}
+ $$\Vec{m}\cdot\Vec{n}=|\Vec{m}|\cdot|\Vec{n}|\cdot\cos\alpha$$
+ $$\Vec{m}\cdot\Vec{n}=1$$
+ $$1=\sqrt{3^2+5^2}\cdot\sqrt{\left(-3\right)^2+2^2}\cdot\cos\alpha$$
+ $$\alpha=\arccos\frac{1}{\sqrt{3^2+5^2}\cdot\sqrt{\left(-3\right)^2+2^2}}$$
+ $$\alpha=\ang{87,273689}$$
+ \vfill\null\columnbreak
+ \begin{tikzpicture}
+ \tkzInit[xmax=5,ymax=5,ymin=-5,xmin=-5]
+ \tkzGrid
+ \tkzDrawXY[>=latex]
+ \tkzDefPoint(3,5){M}
+ \tkzDefPoint(-3,2){N}
+ \tkzDefPoint(0,0){O}
+ \tkzDrawPoints(O,M,N)
+ \tkzShowPointCoord[xlabel=$3$, ylabel=$5$](M)
+ \tkzShowPointCoord[xlabel=$-3$, ylabel=$2$](N)
+ \tkzLabelPoint(M){$M (3,5)$}
+ \tkzLabelPoint[above left](N){$N (-3,2)$}
+ \tkzLabelPoints(O)
+ \tkzDrawSegments[->](O,M O,N)
+ \tkzLabelLine[above left](O,M){$\Vec{m}$}
+ \tkzLabelLine[below left](O,N){$\Vec{n}$}
+ \tkzMarkAngle[mark=none,->](M,O,N)
+ \tkzFillAngle[fill=red!30,opacity=.3](M,O,N)
+ \tkzLabelAngle[pos=1.25](M,O,N){$\alpha$}
+ \end{tikzpicture}
+ \end{multicols}
+ \setcounter{enumi}{621}
+ \item Točke $A(-2,4,1)$, $B(0,3,1)$ in $C(3,-1,0)$ so oglišča trikotnika v prostoru. Izračunaj kot $BCA$.
+ \paragraph{Kot $CAB$:}
+ $$
+ \overline{AC}^2=\overline{CB}^2+\overline{BA}^2-2\cdot\overline{CB}\cdot\overline{BA}\cdot\cos\beta
+ $$
+ $$
+ \sqrt{(-5)^2+(5)^2+(1)^2}^2=\sqrt{(3)^2+(-4)^2+(-1)^2}^2+\sqrt{(2)^2+(-1)^2+(0)^2}^2
+ $$
+ $$
+ -2\cdot\sqrt{(3)^2+(-4)^2+(-1)^2}\cdot\sqrt{(2)^2+(-1)^2+(0)^2}\cdot\cos\beta
+ $$
+ $$
+ 51=26+5-2\sqrt{26}\sqrt{5}\cos\beta
+ $$
+ $$
+ 20=-2\sqrt{26}\sqrt{5}\cos\beta
+ $$
+ $$
+ -10=\sqrt{26}\sqrt{5}\cos\beta
+ $$
+ $$
+ -0,877058=\cos\beta
+ $$
+ $$
+ \beta=\arccos-0,877058=\ang{151,289}
+ $$
+ \paragraph{Kot $BCA$:}
+ $$
+ \overline{AB}^2=\overline{AC}^2+\overline{CB}^2-2\cdot\overline{AC}\cdot\overline{CB}\cdot\cos\gamma
+ $$
+ $$
+ 5=51+26-2\sqrt{51}\sqrt{26}\cos\gamma
+ $$
+ $$
+ 36=\sqrt{51}\sqrt{26}\cos\gamma
+ $$
+ $$
+ \arccos0,988623=\gamma=\ang{8,650931}
+ $$
+ \paragraph{Kot $BAC$:}
+ $$
+ \alpha=\ang{180}-(\ang{151,289}+\ang{8,650931})=\ang{20,059583}
+ $$
+ \setcounter{enumi}{626}
+ \item Kolikšen ostri kot oklepa vektor z abscisno osjo, kolikšen z ordinatno in kolikšen z aplikatno osjo?
+ \begin{tasks}[label=\textbf{\xslalph*)}](2)
+ \task $\Vec{a}=(-3,2,5)$
+ \paragraph{Z abscisno:}
+ $$(-1,0,0)\cdot(-3,2,5)=-1\cdot\sqrt{(-3)^2+2^2+5^2}\cdot\cos\alpha$$
+ $$3=6,164414\cos\alpha$$
+ $$\arccos0,486664=\alpha=\ang{60,878}$$
+ \paragraph{Z ordinatno:}
+ $$(0,1,0)\cdot(-3,2,5)=6,164414\cos\beta$$
+ $$\arccos\frac{2}{6,164414}=\beta=\arccos0,324=\ang{71,068}$$
+ \paragraph{Z aplikatno:}
+ $$\arccos\frac{5}{6,164414}=\gamma=\arccos0,811107=\ang{35,795759915}$$
+ \task $\Vec{b}=2\Vec{i}+3\Vec{k}-\Vec{j}$
+ \paragraph{Z abscisno:}
+ $$(1,0,0)\cdot(2,-1,3)=\sqrt{2^2+(-1)^2+3^2}^2\cdot\cos\delta$$
+ $$\arccos\frac{2}{3,731657}=\delta=\arccos0,534552=\ang{57,688}$$
+ \paragraph{Z ordinatno:}
+ $$\arccos\frac{-1}{-3,731657}=\epsilon=\arccos0,267261=\ang{74,498640}$$
+ \paragraph{Z aplikatno:}
+ $$\arccos\frac{3}{3,731657}=\zeta=\arccos0,801783726=\ang{36,699225}$$
+ % yeet
+ \end{tasks}
+ \setcounter{enumi}{631}
+ \item Dani sta točki $A(3,-1,0)$ in $B(-2,2,2)$. Zapiši enotski vektor $\Vec{e}$ v smeri vektorja $\vektor{AB}$.
+ $$\Vec{e}=\vektor{AB}\cdot\frac{1}{\sqrt{5^2+(-3)^2+(-2)^2}}=0,162221\cdot\vektor{AB}$$
+ \setcounter{enumi}{641}
+ \newpage
+ \item Naj bo $\Vec{a}=(1,k,1)$ in $\Vec{b}=(2-k,1,-k^2)$. Določi:
+ \begin{enumerate}[label=\textbf{\xslalph*)}]
+ \item $k\in\mathbb{R}$ tako, da bosta vektorja $\Vec{a}$ in $\Vec{b}$ pravokotna,
+ $$(1,k,1)\cdot(2-k,1,-k^2)=0=(2-k^2) \rightarrow k=\sqrt{2}$$
+ \item $k\in\mathbb{R}$ tako, da bo vektor $\Vec{a}$ oklepal s ordinatno osjo kot $\ang{60}$,
+ $$
+ (0,1,0)\cdot(1,k,1)=\sqrt{2+k^2}\frac{1}{2}
+ $$
+ $$
+ 2k=\sqrt{2+k^2} \rightarrow 4k^2=2+k^2 \rightarrow 3k^2=2 \rightarrow k^2=\frac{2}{3}
+ $$
+ $$
+ k=\sqrt{\frac{2}{3}}
+ $$
+ \item $k\in\mathbb{N}$ tako, da bo dolžina vektorja $\Vec{a}$ enaka $3\sqrt{3}$,
+ $$
+ 3\sqrt{3}=\sqrt{2+k^2} \rightarrow 25=k^2 \rightarrow k=5
+ $$
+ \item $k\in\mathbb{Z}$ tako, da bo skalarni produkt $\Vec{a}\cdot\Vec{b}$ naravno število.
+ $$(1,k,1)\cdot(2-k,1,-k^2)=2-k+k-k^2=2-k^2 \rightarrow k=1$$
+ \end{enumerate}
+ \setcounter{enumi}{651}
+ \item Točke $A(2,3,4)$, $B(-2,3,2)$, $C(2,0,2)$ so oglišča trikotnika. Določi točko:
+ \begin{enumerate}[label=\textbf{\xslalph*)}]
+ \item $D$ tako, da bo $ABCD$ paralelogram,
+ $$
+ D=C-(B-A)=C-B+A=(2,0,2)-(-2,3,2)+(2,3,4)=(6,0,4)
+ $$
+ \item $E$ na daljici $AB$ tako, da bo $|AE|:|AB|=3:4$,
+ $$
+ E=A+\frac{3}{4}(B-A)=(2,3,4)+\frac{3}{4}(-4,0,-2)=\left(-1,3,2\frac{1}{2}\right)
+ $$
+ \item $F$, ki je težišče trikotnika $ABC$,
+ $$
+ M=A+\frac{1}{2}(B-A)=(2,3,4)+\frac{1}{2}(-4,0,-2)=\left(0,3,3\right)
+ $$
+ $$
+ F=M+\frac{1}{3}(C-M)=(0,3,3)+\frac{1}{3}(2,-3,-1)=\left(\frac{2}{3},2,2\frac{2}{3}\right)
+ $$
+ \item $G$ tako, da bo točka $C$ težišče trikotnika $ABG$.
+ $$
+ G=M+3\cdot(C-M)=(0,3,3)+(6,-9,-3)=(6,-6,0)
+ $$
+ \end{enumerate}
+\end{enumerate}
+\section{Zaključek}
+Ta dokument je informativne narave in se lahko še spreminja. Najnovejša različica, torej PDFji in
+\hologo{LaTeX}\footnote{Za izdelavo dokumenta potrebujete \texttt{TeXLive 2020}.}
+izvorna koda, zgodovina sprememb in prejšnje različice, je na voljo v mojem šolskem Git repozitoriju na
+\url{https://git.sijanec.eu/sijanec/sola-gimb-2} v mapi
+\href{https://git.sijanec/sola-gimb-2/src/branch/master/\predmdn/\predmkaj/\stevilkadn/}{/\predmdn/\predmkaj/\stevilkadn/}. Povezava za ogled zadnje različice tega dokumenta v PDF obliki je \url{http://razor.arnes.si/~asija3/files/sola/gimb/2/\predmdn/\predmkaj/\stevilkadn/dokument.pdf} in/ali \url{https://git.sijanec.eu/sijanec/sola-gimb-2/raw/branch/master/\predmdn/\predmkaj/\stevilkadn/dokument.pdf}.
+\if\razhroscevanje1
+\vfill
+\section*{Razhroščevalne informacije}
+Te informacije so generirane, ker je omogočeno razhroščevanje. Pred objavo dokumenta izklopite razhroščevanje. To naredite tako, da nastavite ukaz \texttt{razhroscevanje} na 0 v začetku dokumenta.
+
+Grafi imajo natančnost \functionSamples\space točk na graf.
+
+Konec generiranja dokumenta: \today\ ob \currenttime\footnote{To ne nakazuje dejanskega časa, ko je bil dokument napisan, temveč čas, ko je bi dokument generiran v PDF/DVI obliko. Isto velja za datum v glavi dokumenta. Če berete direktno iz LaTeX datoteke, bo to vedno današnji datum.}%\input|"date -Ins"
+
+Dokument se je generiral R0qK1KR2 \SI{}{\second}.
+\fi
+% \item $$$$ aaasecgeninsaaa R0qK1KR2
+\end{document}
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