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\documentclass[a4paper,10pt]{article}
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\usepackage{myXsim}
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\usepackage{pgfplots}
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% \pgfplotsset{compat = newest}
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% \usepgfplotslibrary{external}
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% \tikzexternalize
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% Title Page
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\title{DM 3 \hfill \Var{Nom}}
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\tribe{2nd6}
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\date{À rendre pour mardi 25 janvier 2022}
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\xsimsetup{
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solution/print = false
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}
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\begin{document}
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\maketitle
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\begin{exercise}[subtitle={Calculs de fractions}, points=2]
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Détailler les calculs suivants et donner le résultat sous la forme d'une fraction irréductible.
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\begin{multicols}{3}
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\begin{enumerate}[label={\Alph*=}]
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%- set A = rdm.expression("{a} / {b} + {c}", ["a!=b", "b > 1"], global_config={"min_max":(1, 10)})
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\item $\Var{A}$
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%- set B = rdm.expression("{a} / {b} + {c} / {d}", ["a!=b", "b > 1"], global_config={"min_max":(-10, 10), "rejected":[-1, 0, 1]})
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\item $\Var{B}$
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%- set C = rdm.expression("{a} / {b} * {c} / {k*b}", ["a!=b", "c!=b", "b > 1"], global_config={"min_max":(1, 10)})
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\item $\Var{C}$
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\end{enumerate}
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\end{multicols}
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\end{exercise}
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\begin{solution}
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\begin{enumerate}[label={\Alph*=}]
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\item $\Var{A.simplify().explain() | join('=')} = \Var{A.simplify().simplified}$
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\item $\Var{B.simplify().explain() | join('=')} = \Var{B.simplify().simplified}$
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\item $\Var{C.simplify().explain() | join('=')} = \Var{C.simplify().simplified}$
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\end{enumerate}
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\end{solution}
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\begin{exercise}[subtitle={Développer}, points=2]
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Développer puis réduire les expressions suivantes
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\begin{multicols}{3}
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\begin{enumerate}[label={\Alph*=}]
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%- set A = rdm.expression("{a}x({c}x+{d}) - {b}x", [], global_config={"min_max":(1, 10)})
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\item $\Var{A}$
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%- set B = rdm.expression("({a}x+{b})({c}x+{d})", [], )
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\item $\Var{B}$
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%- set C = rdm.expression("({a}x+{b})^2", [], )
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\item $\Var{C}$
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\end{enumerate}
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\end{multicols}
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\end{exercise}
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\begin{solution}
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\begin{enumerate}
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\item
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\begin{align*}
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A &= \Var{A.simplify().explain() | join('\\\\&=')}
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\end{align*}
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\item
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\begin{align*}
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B &= \Var{B.simplify().explain() | join('\\\\&=')}
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\end{align*}
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\item
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\begin{align*}
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C &= \Var{C.simplify().explain() | join('\\\\&=')}
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\end{align*}
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\end{enumerate}
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\end{solution}
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\begin{exercise}[subtitle={Inéquation et tableaux}, points=3]
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Tracer le tableau de signe des fonctions suivantes en le démontrant à l'aide de la résolution d'une inéquation.
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\begin{multicols}{2}
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\begin{enumerate}
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%- set f = rdm.expression("{a}x + {b}", global_config={"min_max":(1, 20)})
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\item $f(x) = \Var{f}$
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%- set g = rdm.expression("{a}x + {b}", global_config={"min_max":(1, 20)})
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\item $g(x) = \Var{g}$
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\end{enumerate}
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\end{multicols}
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\end{exercise}
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\begin{solution}
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\begin{enumerate}
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\item
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Pour déterminer les valeurs de $x$ pour lesquelles $f(x)$ est positive, il faut résoudre l'inéquation
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%- set cst = -f[0]
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%- set coef = f[1]
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%- set racine = cst / coef
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\begin{align*}
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f(x) & \geq 0 \\
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\Var{f} & \geq 0 \\
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\Var{f + cst} &\geq \Var{0 + cst} \\
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\frac{\Var{f + cst}}{\Var{coef}} &\geq \frac{\Var{cst}}{\Var{coef}} \\
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x &\geq \Var{racine.simplify()} \\
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\end{align*}
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Donc $f(x)$ est positif quand $x$ est supérieur à $\Var{racine}$. On en déduit le tableau de signe
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\begin{center}
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\begin{tikzpicture}
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\tkzTabInit[lgt=2,espcl=1]{$ t $/1,$ f(t) $/1}{, $\Var{racine}$ ,}
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\tkzTabLine{, -, z, +, }
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\end{tikzpicture}
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\end{center}
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\item
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Pour déterminer les valeurs de $x$ pour lesquelles $g(x)$ est positive, il faut résoudre l'inéquation
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%- set cst = -g[0]
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%- set coef = g[1]
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%- set racine = cst / coef
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\begin{align*}
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g(x) & \geq 0 \\
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\Var{g} & \geq 0 \\
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\Var{g + cst} &\geq \Var{0 + cst} \\
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\frac{\Var{g + cst}}{\Var{coef}} &\geq \frac{\Var{cst}}{\Var{coef}} \\
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x &\geq \Var{racine.simplify()} \\
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\end{align*}
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Donc $f(x)$ est positif quand $x$ est supérieur à $\Var{racine}$. On en déduit le tableau de signe
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\begin{center}
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\begin{tikzpicture}
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\tkzTabInit[lgt=2,espcl=1]{$ t $/1,$ g(t) $/1}{, $\Var{racine}$ ,}
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\tkzTabLine{, -, z, +, }
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\end{tikzpicture}
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\end{center}
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\end{enumerate}
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\end{solution}
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\begin{exercise}[subtitle={Vecteurs}, points=2]
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\begin{enumerate}
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\item Tracer les vecteurs $\vect{z} = \vect{u} + \vect{v}$ et $\vect{y} = 2\vect{u} - \vect{v}$ (le vecteur peur sortir du cadre)
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%- set xa1, ya1 = rdm.list(["x", "y"], global_config={"min_max": (-5, 5), "rejected":[-2, -1, 0, 1, 2]})
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%- set xa2, ya2 = -xa1, ya1
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%- set aminx = min(0, xa1, xa2, xa1+xa2, 2*xa1-xa2)
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%- set amaxx = max(0, xa1, xa2, xa1+xa2, 2*xa1-xa2)
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%- set aminy = min(0, ya1, ya2, ya1+ya2, 2*ya1-ya2)
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%- set amaxy = max(0, ya1, ya2, ya1+ya2, 2*ya1-ya2)
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\begin{center}
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\begin{tikzpicture}[scale=0.4]
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\draw (\Var{aminx-1}, \Var{aminy-1}) rectangle (\Var{amaxx+1}, \Var{amaxy+1});
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\draw[very thick, ->] (0, 0) -- node [midway, sloped, above] {$\vect{u}$} (\Var{xa1}, \Var{ya1});
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\draw[very thick, -> ] (0, 0) -- node [midway, sloped, above] {$\vect{v}$} (\Var{xa2}, \Var{ya2});
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\end{tikzpicture}
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\end{center}
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\item Tracer la force résultat de la somme des 3 forces exercées sur le point $0$ représenté ci-dessous.
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%- set x1, y1 = rdm.list(["x", "y"], global_config={"min_max": (-5, 5), "rejected":[-2, -1, 0, 1, 2]})
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%- set x2, y2 = -x1, y1
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%- set x3, y3 = x2, 0
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%- set minx = min(0, x1, x2, x3, x1+x2, x2+x3, x1+x3, x1+x2+x3 )
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%- set maxx = max(0, x1, x2, x3, x1+x2, x2+x3, x1+x3, x1+x2+x3 )
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%- set miny = min(0, y1, y2, y3, y1+y2, y2+y3, y1+y3, y1+y2+y3 )
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%- set maxy = max(0, y1, y2, y3, y1+y2, y2+y3, y1+y3, y1+y2+y3 )
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\begin{center}
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\begin{tikzpicture}[scale=0.4]
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\draw (\Var{minx-1}, \Var{miny-1}) rectangle (\Var{maxx+1}, \Var{maxy+1});
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\draw[very thick, ->] (0, 0) -- node [midway, sloped, above] {$\vect{F_1}$} (\Var{x1}, \Var{y1});
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\draw[very thick, -> ] (0, 0) -- node [midway, sloped, above] {$\vect{F_2}$} (\Var{x2}, \Var{y2});
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\draw[very thick, -> ] (0, 0) -- node [midway, sloped, above] {$\vect{F_3}$} (\Var{x3}, \Var{y3});
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\end{tikzpicture}
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\end{center}
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\end{enumerate}
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\end{exercise}
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\begin{solution}
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\begin{enumerate}
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\item
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\begin{center}
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\begin{tikzpicture}[scale=0.4]
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\draw (\Var{aminx-1}, \Var{aminy-1}) rectangle (\Var{amaxx+1}, \Var{amaxy+1});
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\draw[very thick, ->] (0, 0) -- node [midway, sloped, above] {$\vect{u}$} (\Var{xa1}, \Var{ya1});
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\draw[very thick, -> ] (0, 0) -- node [midway, sloped, above] {$\vect{v}$} (\Var{xa2}, \Var{ya2});
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\draw[very thick, ->, color=blue] (0, 0)
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-- ++ (\Var{xa1}, \Var{ya1}) node [midway, sloped, above] {$\vect{u}$}
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-- ++ (\Var{xa2}, \Var{ya2}) node [midway, sloped, above] {$\vect{v}$} ;
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\draw[very thick, ->, color=blue] (0, 0) -- node [midway, sloped, left] {$\vect{z}$} (\Var{xa2+xa1}, \Var{ya2+ya1});
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\draw[very thick, ->, color=green] (0, 0)
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-- ++ (\Var{xa1}, \Var{ya1}) node [midway, sloped, above] {$\vect{u}$}
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-- ++ (\Var{xa1}, \Var{ya1}) node [midway, sloped, above] {$\vect{u}$}
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-- ++ (-\Var{xa2}, -\Var{ya2}) node [midway, sloped, above] {$-\vect{v}$} ;
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\draw[very thick, ->, color=green] (0, 0) -- node [midway, sloped, above] {$\vect{y}$} (\Var{2*xa1 - xa2}, \Var{2*ya1 - ya2});
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\end{tikzpicture}
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\end{center}
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\item
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\begin{center}
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\begin{tikzpicture}[scale=0.4]
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\draw (\Var{minx-1}, \Var{miny-1}) rectangle (\Var{maxx+1}, \Var{maxy+1});
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\draw[very thick, ->] (0, 0) -- node [midway, sloped, above] {$\vect{F_1}$} (\Var{x1}, \Var{y1});
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\draw[very thick, -> ] (0, 0) -- node [midway, sloped, above] {$\vect{F_2}$} (\Var{x2}, \Var{y2});
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\draw[very thick, -> ] (0, 0) -- node [midway, sloped, above] {$\vect{F_3}$} (\Var{x3}, \Var{y3});
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\draw[very thick, ->, color=blue] (0, 0)
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--++ (\Var{x1}, \Var{y1}) node [midway, sloped, above] {$\vect{F_1}$}
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--++ (\Var{x2}, \Var{y2}) node [midway, sloped, above] {$\vect{F_2}$}
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--++ (\Var{x3}, \Var{y3}) node [midway, sloped, above] {$\vect{F_3}$}
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;
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\draw[very thick, ->, color=blue] (0, 0) -- (\Var{x1+x2+x3}, \Var{y1+y2+y3}) node [midway, sloped, above] {$\vect{F_1}+\vect{F_2}+\vect{F_3}$};
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\end{tikzpicture}
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\end{center}
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\end{enumerate}
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\end{solution}
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\end{document}
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