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+\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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