\documentclass[a4paper,slovak]{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amssymb} \usepackage{a4} \usepackage{hyphenat} %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.00.0.2570} %TCIDATA{} %TCIDATA{Created=Thursday, February 21, 2002 14:53:39} %TCIDATA{LastRevised=Monday, October 03, 2005 09:58:42} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{AllPages= %F=36,\PARA{038
Scientific Notebook: On Line Mathematics\dotfill \thepage} %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \section{Vektorov\'{y} po\v{c}et v trojrozmernom priestore} \subsection{D\^{o}kaz vety} Nech $\overline{u}$ je \v{l}ubo\v{l}n\'{y} vektor. Pod\v{l}a predo\v{s}lej vety s\'{u} vektory $\overline{a},\overline{b},\overline{c},\overline{u}$ line\'{a}rne z\'{a}visl\'{e}. Existuj\'{u} teda \v{c}\'{\i}sla $\alpha ,\beta ,\gamma ,\upsilon $, z ktor\'{y}ch aspo\v{n} jedno je nenulov\'{e}, tak, \v{z}e \[ \alpha \overline{a}+\beta \overline{b}+\gamma \overline{c}+\upsilon \overline{u}=\overline{0} \]% Ak by $\upsilon =0$, znamenalo by to, \v{z}e vektory $\overline{a},\overline{% b},\overline{c}$ s\'{u} line\'{a}rne z\'{a}visl\'{e}, \v{c}o nie je pravda. Preto $\upsilon \neq 0$ a m\^{o}\v{z}eme vyjadri\v{t} vektor $\overline{u}$: \[ \overline{u}=-\dfrac{\alpha }{\upsilon }\,\overline{a}-\dfrac{\beta }{% \upsilon }\,\overline{b}-\dfrac{\gamma }{\upsilon }\,\overline{c} \] Predpokladajme, \v{z}e s\'{u} mo\v{z}n\'{e} dve vyjadrenia: \[ \overline{u}=\alpha _{1}\overline{a}+\beta _{1}\overline{b}+\gamma _{1}% \overline{c} \]% \[ \overline{u}=\alpha _{2}\overline{a}+\beta _{2}\overline{b}+\gamma _{2}% \overline{c}. \]% Potom \[ \overline{0}=\overline{u}-\overline{u}=(\alpha _{1}-\alpha _{2})\overline{a}% +(\beta _{1}-\beta _{2})\overline{b}+(\gamma _{1}-\gamma _{2})\overline{c} \]% Z line\'{a}rnej nez\'{a}vislosti vektorov $\overline{a},\overline{b},% \overline{c}$ vypl\'{y}va \[ \alpha _{1}-\alpha _{2}=0,\;\beta _{1}-\beta _{2}=0,\;\gamma _{1}-\gamma _{2}=0 \]% odkia\v{l} \[ \alpha _{1}=\alpha _{2},\;\beta _{1}=\beta _{2},\;\gamma _{1}=\gamma _{2}.\blacksquare \] \begin{center} \begin{tabular}{|c|} \hline \textbf{% %TCIMACRO{\hyperref{Sp\"{a}\v{t}}{}{}{A61.tex#6}}% %BeginExpansion \msihyperref{Sp\"{a}\v{t}}{}{}{A61.tex#6}% %EndExpansion } \\ \hline \end{tabular} \end{center} \end{document}