% !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}
