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diff --git a/mat/domace_naloge/20/dokument.tex b/mat/domace_naloge/20/dokument.tex new file mode 100644 index 0000000..d366b2d --- /dev/null +++ b/mat/domace_naloge/20/dokument.tex @@ -0,0 +1,289 @@ +% !TeX encoding = UTF-8 +% !TeX spellcheck = sl_SI +% do-vimlatex-onwrite +\documentclass[]{article} +\usepackage[utf8]{inputenc} +\usepackage{etoolbox} +\usepackage[hidelinks]{hyperref} +\usepackage[a4paper, total={7in, 10in}]{geometry} +\usepackage{graphicx} +\usepackage{hologo} +\usepackage{amssymb} +\usepackage{calculator} +\usepackage{pgfplots} +\usepackage[inline]{enumitem} +\usepackage{siunitx} +\usepackage{multicol} +\usepackage{tabularcalc} +\usepackage{amsmath} +\usepackage{float} +\usepackage{tasks} +\usepackage{filecontents} +\usepackage{textcomp} +\usepackage{tkz-euclide} +\usepackage{dirtytalk} +\usepackage{csquotes} +\usepackage{listings} +\usepackage{datetime} % [ddMMyyyy] +\usepackage{ccicons} +\usepackage{chemformula} +\usepackage{gensymb} +\usepackage[normalem]{ulem} +\usepackage[slovene]{babel} +\usepackage{cancel} +\usepackage{tabularx} +\usepackage{xcolor} +\usepackage{colortbl} +\usepackage{exsheets} +\usepackage{harpoon} +\newcolumntype{Y}{>{\centering\arraybackslash}X} +% \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 +\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}{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]{% + \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{$#2$}% +\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} |