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diff --git a/mat/domace_naloge/22/dokument.tex b/mat/domace_naloge/22/dokument.tex new file mode 100644 index 0000000..02c59cc --- /dev/null +++ b/mat/domace_naloge/22/dokument.tex @@ -0,0 +1,239 @@ +% !TeX encoding = UTF-8 +% !TeX spellcheck = sl_SI +% do-vimlatex-onwrite +\documentclass[]{article} +\usepackage[utf8]{inputenc} +\usepackage{etoolbox} +\usepackage[hidelinks]{hyperref} +\usepackage{xurl} % da breaka url +\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} +\usepackage{readarray} +\usepackage{forloop} +\usepackage[nomessages]{fp} +\usepackage{multirow} +\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}{Računanje s komponentami } +\newcommand{\predmdn}{mat} +\newcommand{\predmkaj}{domace\_naloge} +\newcommand{\stevilkadn}{22} +\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} +%\tableofcontents +\begin{enumerate}[label=\textbf{\arabic*.}] + \setcounter{enumi}{12} + \item Določi koordinate točke $M$, tako da bo točka $Z(-7, 8, -10)$ razpolovišče daljice $LM$, če je $L(-4, 2, -8)$. + \begin{center} + skica: + \begin{tikzpicture} + \tkzDefPoint(0,0){L} \tkzDefPoint(2,0){Z} \tkzDefPoint(4,0){M} + \tkzDrawLine(L,M) + \tkzDrawPoints(L,Z,M) + \tkzLabelPoints(L,Z,M) + \end{tikzpicture} + \end{center} + $$ + Z\left(-7, 8, -10\right) = Z\left(\frac{-4+M_1}{2}, \frac{2+M_2}{2}, \frac{-8+M_3}{2}\right) + $$ + $$ + \frac{-4+M_1}{2} = -7 \wedge + \frac{2+M_2}{2} = 8 \wedge + \frac{-8+M_3}{2} = -10 + \rightarrow M\left(M_1, M_2, M_3\right) = M\left(-10, 14, -12\right) + $$ + \paragraph{Rešitev} $M\left(-10, 14, -12\right)$ + \item Dani sta točki $A\left(4, -7, -1\right)$ in $B\left(-4, -3, -8\right)$. Določi točko $C$, tako da bo točka $T\left(1, -1, 3\right)$ težišče trikotnika $ABC$. + \begin{center} + skica: + \begin{tikzpicture} + \tkzDefPoint(0,0){A} \tkzDefPoint(4,0){B} \tkzDefPoint(3,3){C} + \tkzDefMidPoint(A,B) \tkzGetPoint{M} \tkzDefMidPoint(B,C) \tkzGetPoint{N} \tkzDefMidPoint(C,A) \tkzGetPoint{O} + \tkzDrawPolygon[fill=red!30, opacity=.3](A,B,C) + \tkzDrawLine(M,C) \tkzDrawLine(N,A) \tkzDrawLine(O,B) + \tkzInterLL(N,A)(O,B) \tkzGetPoint{T} + \tkzDrawPoints(A,B,C,T,M) \tkzLabelPoints(A,B,C,T,M) + \end{tikzpicture} + \end{center} + $$ + M = M\left(\frac{4+\left(-4\right)}{2}, \frac{-7+\left(-3\right)}{2}, \frac{-1+\left(-8\right)}{2}\right) = M\left(0, -5, 4,5\right) + $$ + Vemo, da težišče razpolovi težiščnico na tri enake dele. Naredimo vektor $\vektor{MT}$ in ga pomnožimo s skalarjem tri, da dobimo vektor $\vektor{MC}$. + $$ + \vektor{MT} = \Vec{T} - \Vec{M} = \left(1-0, -1-\left(-5\right), 3-4,5\right) = \left(1, 4, -1,5\right) \rightarrow \vektor{MC} = 3\vektor{MT} = \left(3, 12, -4,5\right) + $$ + $$ + \rightarrow r_C = r_M + \vektor{DC} = \left(0+3, -5+12, 4,5+\left(-4,5\right)\right) = C\left(3, 7, 0\right) + $$ + \parahraph{Rešitev} $C\left(3, 7, 0\right)$ + \item Točka $P$ leži na daljici $AB$ tako, da je $|AP|:|PB|=1:1$. Zapiši koordinate točke $P$, če je $A\left(-6, -13, 2\right)$ in $B\left(-13, 5, 4\right)$. Koordinate zaokroži na dve decimalni mesti. + \begin{center} + skica: + \begin{tikzpicture} + \tkzDefPoint(0,0){A} \tkzDefPoint(2,0){P} \tkzDefPoint(4,0){B} + \tkzDrawLine(A,B) + \tkzDrawPoints(A,P,B) + \tkzLabelPoints(A,P,B) + \end{tikzpicture} + \end{center} + $$ + P = \left(\frac{-6+\left(-13\right)}{2}, \frac{-13+5}{2}, \frac{2+4}{2}\right) = \left(-9,5, -4, 3\right) + $$ + \paragraph{Rešitev} Točka $P$ ima koordinate $P \left(-9,5, -4, 3\right)$ + \item Dane so točke $A\left(6, 5, 2\right)$, $B\left(6, 2, -5\right)$, $C\left(-7, -2, -6\right)$. Določi koordinate točke $D$, tako da bo štirikotnik $ABCD$ paralelogram. + \begin{center} + skica: + \begin{tikzpicture} + \tkzDefPoint(0,0){A} \tkzDefPoint(2,0){B} \tkzDefPoint(3,2){C} \tkzDefPoint(1,2){D} + \tkzDrawPolygon[fill=red!30, opacity=0.3](A,B,C,D) + \tkzDrawPoints(A,B,C,D) \tkzLabelPoints(A,B,C,D) + \end{tikzpicture} + $$ + \vektor{AD} = \vektor{BC} = \Vec{C}-\Vec{B} = \left(-7-6, -2-2, -6-\left(-5\right)\right) = \left(-13, -4, -1\right) + $$ + $$ + \rightarrow r_D = r_A + \vektor{AD} = (-13+6, -4+5, -1+2) = (-7, 1, 1) + $$ + \end{center} + \paragraph{Rešitev} $D\left(-7, 1, 1\right)$ + \item Zapiši prvo komponento vektorja $\Vec{b}=\left(x, -4\right)$, da bo vzporeden z vektorjem $\Vec{a}=\left(6, -2\right)$. + $$ + \Vec{b} = k\cdot\Vec{a} = (6k, -2k) \wedge -2 \cdot 2 = -4 \rightarrow k = -4 \rightarrow \Vec{b} = (12, -4) + $$ + \paragraph{Rešitev} $x = 12$ +\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} |