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diff --git a/mat/domace_naloge/28/dokument.tex b/mat/domace_naloge/28/dokument.tex new file mode 100644 index 0000000..a9fd8b4 --- /dev/null +++ b/mat/domace_naloge/28/dokument.tex @@ -0,0 +1,335 @@ +% !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}{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} |