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Diffstat (limited to 'mat/domace_naloge/20/dokument.tex')
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diff --git a/mat/domace_naloge/20/dokument.tex b/mat/domace_naloge/20/dokument.tex index bf0f096..ee74c49 100644 --- a/mat/domace_naloge/20/dokument.tex +++ b/mat/domace_naloge/20/dokument.tex @@ -134,39 +134,137 @@ Ta dokument vsebuje domačo nalogo, ki zajema snov \textit{\snovdn}pri matematik \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 - \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 $\displaystyle\frac{1}{3}\vektor{a}$ % a - \task $\displaystyle\frac{4}{5}\vektor{a}$ % b - \task $\displaystyle-\frac{3}{7}\vektor{a}$ % c - \task $\displaystyle-\frac{3}{4}\vektor{a}$ % č - \task $\displaystyle\frac{5}{4}\vektor{a}$ % d - \task $\displaystyle\frac{7}{5}\vektor{a}$ % e - \task $\displaystyle-\frac{3}{2}\vektor{a}$ % f - \task $\displaystyle-\frac{8}{5}\vektor{a}$ % g - \task $\displaystyle\sqrt{2}\vektor{a}$ % h - \task $\displaystyle\sqrt{5}\vektor{a}$ % i - \task $\displaystyle-\sqrt{7}\vektor{a}$ % j - \task $\displaystyle-\sqrt{10}\vektor{a}$ % k - \task $\displaystyle\frac{\sqrt{3}}{4}\vektor{a}$ % l - \task $\displaystyle\frac{\sqrt{6}}{7}\vektor{a}$ % m - \task $\displaystyle-\frac{\sqrt{12}}{3}\vektor{a}$ % n - \task $\displaystyle-\frac{\sqrt{15}}{8}\vektor{a}$ % o - \end{primerTasks} + \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} - \tkzDefPoint(0,0){A} \tkzDefPoint(0,3){B} - \tkzDrawPoints(A,B) + \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} - \setcounter{enumi}{319} + \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 $\vektor{m}=2\vektor{a}+3\vektor{b}$ in $\vektor{n}=\vektor{a}-2\vektor{b}$. Izrazi vektor $2\vektor{m}-\frac{1}{2}\vektor{n}$ z vektorjema $\vektor{a}$ in $\vektor{b}$. + \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} |