path: root/mat/domace_naloge/20/dokument.tex

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\usepackage{tasks}
\usepackage{filecontents}
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\usepackage{tkz-euclide}
\usepackage{dirtytalk}
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\usepackage{listings}
\usepackage{datetime} % [ddMMyyyy]
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% \usepackage{multienum} % weird with labels
\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
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		{\LARGE \@title \par}%
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		{\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}{Množenje vektorja s številom }
\newcommand{\predmdn}{mat}
\newcommand{\predmkaj}{domace\_naloge}
\newcommand{\stevilkadn}{20}
\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]{%
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	\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{$#2$}%
\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 domačo nalogo, ki zajema snov \textit{\snovdn}pri matematiki, in njene rešitve, ki sem jih spisal sam. Kjer je bilo potrebno izbrati neke poljubne naloge, sem jih vedno izbral naključno.
\end{abstract}
\tableofcontents
\section{Vaje \textit{Matematika 2}: stran 50, dve sodi in dve lihi nalogi 317-336}
\begin{enumerate}[label=\textbf{\arabic*.}] % 318, 329, 320, 325
	\if\false1
		\setcounter{enumi}{317}
		\item Nariši vektor $\vektor{a}$, dolg \cm{3}. Nariši še vektorje:
			\footnote{Za sezname ne uporabljam več \texttt{enumerate}, ampak \texttt{tasks} (Deutsche Kvalität)}
			\begin{primerTasks}(8)
				\task $\displaystyle\frac{1}{3}\Vec{a}$ % a
				\task $\displaystyle\frac{4}{5}\Vec{a}$ % b
				\task $\displaystyle-\frac{3}{7}\Vec{a}$ % c
				\task $\displaystyle-\frac{3}{4}\Vec{a}$ % č
				\task $\displaystyle\frac{5}{4}\Vec{a}$ % d
				\task $\displaystyle\frac{7}{5}\Vec{a}$ % e
				\task $\displaystyle-\frac{3}{2}\Vec{a}$ % f
				\task $\displaystyle-\frac{8}{5}\Vec{a}$ % g
				\task $\displaystyle\sqrt{2}\Vec{a}$ % h
				\task $\displaystyle\sqrt{5}\Vec{a}$ % i
				\task $\displaystyle-\sqrt{7}\Vec{a}$ % j
				\task $\displaystyle-\sqrt{10}\Vec{a}$ % k
				\task $\displaystyle\frac{\sqrt{3}}{4}\Vec{a}$ % l
				\task $\displaystyle\frac{\sqrt{6}}{7}\Vec{a}$ % m
				\task $\displaystyle-\frac{\sqrt{12}}{3}\Vec{a}$ % n
				\task $\displaystyle-\frac{\sqrt{15}}{8}\Vec{a}$ % o
			\end{primerTasks}
			\begin{center}
				\begin{tikzpicture}[vect/.style={->,shorten >=3pt,>=latex'}]
					% 0	
					\tkzDefPoint(0,0){A} \tkzDefPoint(3,0){B}
					\tkzDrawPoints[red](A,B) \tkzDrawSegments[vect](A,B)
					\tkzLabelLine[above](A,B){$\Vec{a}$}
					% a
					\tkzDefPoint(30:3){aA}
					\tkzDrawLine(A,aA) \tkzDrawLine(aA,B)
					\tkzDefPoint(1,0){a1} \tkzDefPoint(30:1){a2} \tkzDefPoint(30:2){a3}
					\tkzDefPoint(2,0){a4}
					\tkzCompass(A,a2) \tkzCompass(a2,a3) \tkzDrawLine(a3,a4)
					\tkzDrawLine(a1,a2) \tkzDrawPoints[red](a1) \tkzDrawSegment[vect](A,a1)
					\tkzLabelLine[below](A,a1){$\frac{1}{3}\Vec{a}$}
					% b
					\tkzDefPoint(30:(3/5)){b1} \tkzDefPoint(30:(3/5)*2){b2}
					\tkzDefPoint(30:(3/5)*3){b3} \tkzDefPoint(30:(3/5)*4){b4}
					\tkzCompass(A,b1) \tkzCompass(b1,b2) \tkzCompass(b2,b3)
					\tkzCompass(b3,b4) \tkzCompass(b4,aA)
					\tkzDefPoint((3/5)*4,0){b5} \tkzDrawLine(b4,b5) \tkzDrawPoints[red](b5)
					\tkzDrawSegments[vect](A,b5)
					\tkzLabelLine[below](A,b5){$\frac{4}{5}\Vec{a}$}
				\end{tikzpicture}
			\end{center}
		\fi
	\setcounter{enumi}{318}
	\item Nariši trikotnik $ABC$ s podatki $a=\cm{4}$, $b=\cm{6}$, $c=\cm{7}$. Nariši še $\displaystyle\frac{1}{3}\vektor{AC}+\sqrt{2}\vektor{AB}$. % 319
		\begin{center}
			\begin{tikzpicture}[vect/.style={->,shorten >=3pt,>=latex'}]
				\tkzDefPoint(0,0){A} \tkzDefPoint(7,0){B}
				\tkzDefPoint(6,0){bh} \tkzDefPoint(7-4,0){ah}
				\tkzDefPoint(0,6){3h} \tkzDefPoint(0,4){2h} \tkzDefPoint(0,2){1h}
				\tkzCompass(A,B) \tkzDrawLine(A,B) % \tkzLabelLine[below](A,B){$c$}
				\tkzInterCC(A,bh)(B,ah) \tkzGetPoints{C}{CX}
				\tkzDrawPolygon[fill=red!30, opacity=.3](A,B,C)
				\tkzLabelLine[below right](A,C){$\vektor{AC}$}
				\tkzDrawLine(A,3h) \tkzCompass(A,1h) \tkzCompass(1h,2h)
				\tkzCompass(2h,3h) \tkzDrawLine(3h,C)
				\tkzDefPointWith[colinear=at 1h](3h,C) \tkzGetPoint{Z}
				\tkzInterLL(A,C)(1h,Z) \tkzGetPoint{T}
				\tkzLabelLine[above](A,T){$\frac{1}{3}\vektor{AC}$}
				% sedaj pa sqrt(2)AB
				\tkzDefPoint(0,7){P} \tkzCompass(A,P)
				\tkzDefPoint(7,7){K} \tkzDrawLine(B,K) \tkzDrawLine(P,K)
				\tkzInterLC(A,B)(A,K) \tkzGetPoints{K2X}{K2} \tkzCompass(A,K2)
				\tkzLabelLine[below](A,K2){$\sqrt{2}\vektor{AB}$}
				% sedaj pa seštevek
				\tkzDefPointWith[colinear=at B](A,C) \tkzGetPoint{S}
				\tkzLabelLine[above left](A,S){$\frac{1}{3}\vektor{AC}+\sqrt{AB}$}
				\tkzDrawSegments[vect](A,C A,T, A,K A,K2)
				\tkzDrawSegments[red, vect](A,S)
				\tkzMarkRightAngles(A,P,K P,K,B K,B,A B,A,P)
				\tkzDrawPoints[red](A,B,C,T,K,K2,S) \tkzLabelPoints(A,B,C)
			\end{tikzpicture}
		\end{center}
	\newpage
	\item Nariši pravokotnik $ABCD$ s podatki: $a=\cm{6}$, $b=\cm{2}$. Nariši še $\displaystyle-\frac{3}{2}\vektor{AC}+\frac{\sqrt{2}}{2}\vektor{BC}$.
		\begin{center}
			\begin{tikzpicture}[vect/.style={->,shorten >=3pt,>=latex'}]
				\tkzDefPoint(0,0){A} \tkzDefPoint(6,2){C} \tkzDefPoint(6,0){B}
				\tkzDefPoint(0,2){D}
				\tkzDrawPolygon[fill=red!30, opacity=.3](A,B,C,D)
				\tkzDefPoint(0,(2/3)){a1} \tkzDefPoint(0,(2/3)*2){a2}
				\tkzCompass(A,a1) \tkzCompass(a1,a2) \tkzCompass(a2,D)
				\tkzDefPointWith[colinear=at a1](D,C) \tkzGetPoint{Z}
				\tkzInterLL(A,C)(a1,Z) \tkzGetPoint{V}
				\tkzDrawLine(a1,V) \tkzLabelLine[above left](A,C){$\vektor{AC}$}
				\tkzLabelLine[below](V,C){$-\frac{2}{3}\vektor{AC}$}
				% zdaj pa še koren iz dve polovic vektorja BC
				\tkzDefPoint(6-2,0){K} \tkzDefPoint(6-2,2){P}
				\tkzCompass(B,K) \tkzCompass(C,P)
				\tkzInterLC(C,B)(C,K) \tkzGetPoints{K2X}{K2}
				\tkzDefMidPoint(C,K2) \tkzGetPoint{K22}
				\tkzLabelLine[left](C,K22){$\frac{\sqrt{2}}{2}\vektor{CB}$}
				\tkzDefPointWith[colinear=at K22](C,V) \tkzGetPoint{S}
				\tkzDrawPoints[red](A,B,C,D,V,K2,K22,S) \tkzLabelPoints[](A,B,C,D)
				\tkzDrawSegments[vect](A,C C,V C,K2 C,K22)
				\tkzDrawSegments[vect, red](C,S)
				\tkzMarkRightAngles(C,P,K P,K,B K,B,C B,C,P)
			\end{tikzpicture}
		\end{center}
	\setcounter{enumi}{324}
	\item Naj bo $\Vec{m}=2\Vec{a}+3\Vec{b}$ in $\Vec{n}=\Vec{a}-2\Vec{b}$. Izrazi vektor $2\Vec{m}-\frac{1}{2}\Vec{n}$ z vektorjema $\Vec{a}$ in $\Vec{b}$.
		$$
			2\Vec{a}+3\Vec{b}-(\frac{1}{2}(\Vec{a}-2\Vec{b})) = 
			2\Vec{a} + 3\Vec{b} - \frac{1}{2}\Vec{a} + \frac{1}{2} \cdot 2\Vec{b} =
			1\frac{1}{2}\Vec{a}+4\Vec{b}
		$$
	\setcounter{enumi}{328}
	\item Nariši daljico $AB$, dolgo \cm{6}. Na njen nariši tako točno $M$, da velja $|AM|:|MB|=1:2$, in tako točno $N$, da velja $|AN|:|AB|=1:2$. Vektorja $\vektor{AM}$ in $\vektor{AN}$ izrazi z vektorjem $\vektor{AB}$.
		\begin{center}
			\begin{tikzpicture}[vect/.style={->,shorten >=3pt,>=latex'}]
				\tkzDefPoint(0,0){A} \tkzDefPoint(6,0){B}
				\tkzCompass(A,B) \tkzDrawLine(A,B) % \tkzLabelLine[below](A,B){$AB$}
				\tkzDefPoint(30:6){3h} \tkzDefPoint(30:4){2h} \tkzDefPoint(30:2){1h}
				\tkzDrawLine(A,3h) \tkzDrawLine(3h,B)
				\tkzCompass(A,1h) \tkzCompass(1h,2h) \tkzCompass(2h,3h)
				\tkzDefPoint((6/3)*1,0){M} \tkzDrawLine(h1,M)
				\tkzDefMidPoint(A,B) \tkzGetPoint{N}
				\tkzDrawPoints[red](A,B,M,N) \tkzLabelPoints[below left](A,B,M,N)
				\tkzDrawSegments[vect](A,M A,N)
			\end{tikzpicture}
		\end{center}
		$$
			\vektor{AM} = \frac{1}{3}\vektor{AB}
		$$
		$$
			\vektor{AN} = \frac{1}{2}\vektor{AB}
		$$
\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
\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}