458 lines
8.3 KiB
TeX
458 lines
8.3 KiB
TeX
\documentclass{article}
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\usepackage{mathtools}
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\usepackage{amssymb}
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\usepackage[makeroom]{cancel}
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\title{Mécanique du point - TD 1 \\ NON CORRIGÉ}
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\author{Timéo Pochin}
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\begin{document}
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\maketitle
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\section*{Exercice 1}
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\subsection*{a)}
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\begin{align}
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\nonumber
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\vec{u}\land\vec{v}
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&=
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\begin{pmatrix}
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1 \\
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2 \\
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3
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\end{pmatrix}
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\land
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\begin{pmatrix}
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4 \\
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5 \\
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6
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\end{pmatrix}
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\\ \nonumber
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&=
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\begin{pmatrix}
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2\cdot 6-5\cdot 3 \\
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4\cdot 3-1\cdot 6 \\
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1\cdot 5-4\cdot 2
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\end{pmatrix}
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\\ \nonumber
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&=
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\begin{pmatrix}
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-3 \\
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6 \\
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-3
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\end{pmatrix}
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\\ \nonumber
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&=-3\vec{e_x}+6\vec{e_y}-3\vec{e_z}
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\end{align}
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\subsection*{b)}
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\begin{align}
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\nonumber
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\vec{v}\land\vec{u}
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&=
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\begin{pmatrix}
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4 \\
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5 \\
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6
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\end{pmatrix}
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\land
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\begin{pmatrix}
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1 \\
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2 \\
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3
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\end{pmatrix}
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\\ \nonumber
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&=
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\begin{pmatrix}
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5\cdot 3-2\cdot 6 \\
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1\cdot 6-4\cdot 3 \\
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4\cdot 2-1\cdot 5
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\end{pmatrix}
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\\ \nonumber
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&=
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\begin{pmatrix}
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3 \\
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-6 \\
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3
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\end{pmatrix}
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\\ \nonumber
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&=3\vec{e_x}-6\vec{e_y}+3\vec{e_z}
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\end{align}
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\subsection*{c)}
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\begin{align}
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\nonumber
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\vec{u}\cdot\vec{v}
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&=
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\begin{pmatrix}
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1 \\
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2 \\
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3
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\end{pmatrix}
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\cdot
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\begin{pmatrix}
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4 \\
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5 \\
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6
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\end{pmatrix}
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\\ \nonumber
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&=1\cdot 4+2\cdot 5+3\cdot 6\\ \nonumber
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&=32
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\end{align}
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\subsection*{d)}
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\begin{align}
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\nonumber
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\vec{u}\cdot(\vec{u}\land\vec{v})
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&=
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\begin{pmatrix}
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1 \\
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2 \\
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3
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\end{pmatrix}
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\cdot
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\begin{pmatrix}
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-3 \\
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6 \\
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-3
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\end{pmatrix}
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\\ \nonumber
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&=1\cdot-3+2\cdot 6+3\cdot -3\\ \nonumber
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&=0
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\end{align}
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\subsection*{e)}
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\begin{align}
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\nonumber
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\|\vec{u}\|
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&=\sqrt{1^2+2^2+3^2}
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\\ \nonumber
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&=\sqrt{14}
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\end{align}
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\subsection*{f)}
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\begin{align}
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\nonumber
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\|\vec{u}+\vec{v}\|
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&=
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\left\|
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\begin{pmatrix}
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1 \\
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2 \\
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3
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\end{pmatrix}
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+
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\begin{pmatrix}
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4 \\
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5 \\
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6
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\end{pmatrix}
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\right\|
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\\ \nonumber
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&=
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\left\|
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\begin{pmatrix}
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5 \\
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7 \\
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9
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\end{pmatrix}
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\right\|
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\\ \nonumber
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&=\sqrt{5^2+7^2+9^2}
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\\ \nonumber
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&=\sqrt{155}
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\end{align}
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\subsection*{g)}
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\begin{align}
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\nonumber
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(\vec{u}\land\vec{v})\land\vec{w}
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&=
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\begin{pmatrix}
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-3 \\
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6 \\
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-3
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\end{pmatrix}
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\land
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\begin{pmatrix}
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7 \\
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8 \\
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9
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\end{pmatrix}
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\\ \nonumber
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&=
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\begin{pmatrix}
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6\cdot 9-8\cdot-3 \\
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7\cdot-3-9\cdot-3 \\
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-3\cdot 8-7\cdot 6
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\end{pmatrix}
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\\ \nonumber
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&=
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\begin{pmatrix}
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78 \\
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6 \\
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-66
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\end{pmatrix}
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\\ \nonumber
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&=78\vec{e_x}+6\vec{e_y}-66\vec{e_z}
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\end{align}
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\subsection*{h)}
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\begin{align}
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\nonumber
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\vec{u}\land(\vec{v}\land\vec{w})
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&=
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\begin{pmatrix}
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1 \\
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2 \\
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3
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\end{pmatrix}
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\land
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\left(
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\begin{pmatrix}
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4 \\
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5 \\
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6
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\end{pmatrix}
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\land
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\begin{pmatrix}
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7 \\
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8 \\
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9
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\end{pmatrix}
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\right)
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\\ \nonumber
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&=
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\begin{pmatrix}
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1 \\
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2 \\
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3
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\end{pmatrix}
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\land
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\begin{pmatrix}
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5\cdot 9-8\cdot 6 \\
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7\cdot 6-4\cdot 9 \\
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4\cdot 8-7\cdot 5
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\end{pmatrix}
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\\ \nonumber
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&=
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\begin{pmatrix}
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1 \\
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2 \\
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3
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\end{pmatrix}
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\land
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\begin{pmatrix}
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-3 \\
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6 \\
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-3
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\end{pmatrix}
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\\ \nonumber
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&=
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\begin{pmatrix}
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2\cdot-3-6\cdot 3 \\
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-3\cdot 3-1\cdot-3 \\
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1\cdot 6--3\cdot 3
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\end{pmatrix}
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\\ \nonumber
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&=
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\begin{pmatrix}
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-24 \\
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-6 \\
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12
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\end{pmatrix}
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\\ \nonumber
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&= -24\vec{e_x}-6\vec{e_y}+12\vec{e_z}
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\end{align}
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\pagebreak
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\section*{Exercice 2}
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\begin{align}
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\nonumber
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\vec{u}\land(\vec{v}\land\vec{w})
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&=
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\begin{pmatrix}
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u_x \\
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u_y \\
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u_z
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\end{pmatrix}
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\land
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\begin{pmatrix}
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v_x \\
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v_y \\
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v_z
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\end{pmatrix}
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\land
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\begin{pmatrix}
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w_x \\
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w_y \\
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w_z
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\end{pmatrix}
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\\ \nonumber
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&=
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\begin{pmatrix}
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u_x \\
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u_y \\
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u_z
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\end{pmatrix}
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\land
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\begin{pmatrix}
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v_yw_z-v_zw_y \\
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v_zw_x-v_xw_z \\
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v_xw_y-v_yw_x
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\end{pmatrix}
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\\ \nonumber
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&=
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\begin{pmatrix}
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u_x \\
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u_y \\
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u_z
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\end{pmatrix}
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\land
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\begin{pmatrix}
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v_yw_z-v_zw_y \\
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v_zw_x-v_xw_z \\
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v_xw_y-v_yw_x
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\end{pmatrix}
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\\ \nonumber
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&=
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\begin{pmatrix}
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u_y(v_xw_y-v_yw_x)-u_z(v_zw_x-v_xw_z) \\
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u_z(v_yw_z-v_zw_y)-u_x(v_xw_y-v_yw_x) \\
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u_x(v_zw_x-v_xw_z)-u_y(v_yw_z-v_zw_y)
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\end{pmatrix}
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\\
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&=
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\begin{pmatrix}
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u_yv_xw_y-u_yv_yw_x-u_zv_zw_x+u_zv_xw_z \\
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u_zv_yw_z-u_zv_zw_y-u_xv_xw_y+u_xv_yw_x \\
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u_xv_zw_x-u_xv_xw_z-u_yv_yw_z+u_yv_zw_y
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\end{pmatrix}
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\end{align}
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\begin{align}
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\nonumber
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(\vec{u}\cdot\vec{w})\vec{v}-(\vec{u}\cdot\vec{v})\vec{w}
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&=
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\left(
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\begin{pmatrix}
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u_x \\
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u_y \\
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u_z
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\end{pmatrix}
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\cdot
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\begin{pmatrix}
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w_x \\
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w_y \\
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w_z
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\end{pmatrix}
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\right)
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\begin{pmatrix}
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v_x \\
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v_y \\
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v_z
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\end{pmatrix}
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-
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\left(
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\begin{pmatrix}
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u_x \\
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u_y \\
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u_z
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\end{pmatrix}
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\cdot
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\begin{pmatrix}
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v_x \\
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v_y \\
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v_z
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\end{pmatrix}
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\right)
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\begin{pmatrix}
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w_x \\
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w_y \\
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w_z
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\end{pmatrix}
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\\ \nonumber
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&=
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(u_xw_x+u_yw_y+u_zw_z)
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\begin{pmatrix}
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v_x \\
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v_y \\
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v_z
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\end{pmatrix}
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-
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(u_xv_x+u_yv_y+u_zv_z)
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\begin{pmatrix}
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w_x \\
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w_y \\
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w_z
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\end{pmatrix}
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\\ \nonumber
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&=
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\begin{pmatrix}
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(u_xw_x+u_yw_y+u_zw_z)v_x \\
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(u_xw_x+u_yw_y+u_zw_z)v_y \\
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(u_xw_x+u_yw_y+u_zw_z)v_z
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\end{pmatrix}
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-
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\begin{pmatrix}
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(u_xv_x+u_yv_y+u_zv_z)w_x \\
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(u_xv_x+u_yv_y+u_zv_z)w_y \\
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(u_xv_x+u_yv_y+u_zv_z)w_z
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\end{pmatrix}
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\\ \nonumber
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&=
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\begin{pmatrix}
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u_xv_xw_x+u_yv_xw_y+u_zv_xw_z \\
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u_xv_yw_x+u_yv_yw_y+u_zv_yw_z \\
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u_xv_zw_x+u_yv_zw_y+u_zv_zw_z
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\end{pmatrix}
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-
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\begin{pmatrix}
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u_xv_xw_x+u_yv_yw_x+u_zv_zw_x \\
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u_xv_xw_y+u_yv_yw_y+u_zv_zw_y \\
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u_xv_xw_z+u_yv_yw_z+u_zv_zw_z
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\end{pmatrix}
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\\ \nonumber
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&=
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\begin{pmatrix}
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u_xv_xw_x+u_yv_xw_y+u_zv_xw_z-
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(u_xv_xw_x+u_yv_yw_x+u_zv_zw_x) \\
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u_xv_yw_x+u_yv_yw_y+u_zv_yw_z-
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(u_xv_xw_y+u_yv_yw_y+u_zv_zw_y) \\
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u_xv_zw_x+u_yv_zw_y+u_zv_zw_z-
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(u_xv_xw_z+u_yv_yw_z+u_zv_zw_z)
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\end{pmatrix}
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\\ \nonumber
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&=
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\begin{pmatrix}
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u_xv_xw_x+u_yv_xw_y+u_zv_xw_z-
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u_xv_xw_x-u_yv_yw_x-u_zv_zw_x \\
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u_xv_yw_x+u_yv_yw_y+u_zv_yw_z-
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u_xv_xw_y-u_yv_yw_y-u_zv_zw_y \\
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u_xv_zw_x+u_yv_zw_y+u_zv_zw_z-
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u_xv_xw_z-u_yv_yw_z-u_zv_zw_z
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\end{pmatrix}
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\\ \nonumber
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&=
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\begin{pmatrix}
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\cancel{u_xv_xw_x}+u_yv_xw_y+u_zv_xw_z-
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\cancel{u_xv_xw_x}-u_yv_yw_x-u_zv_zw_x \\
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u_xv_yw_x+\cancel{u_yv_yw_y}+u_zv_yw_z-
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u_xv_xw_y-\cancel{u_yv_yw_y}-u_zv_zw_y \\
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u_xv_zw_x+u_yv_zw_y+\cancel{u_zv_zw_z}-
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u_xv_xw_z-u_yv_yw_z-\cancel{u_zv_zw_z}
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\end{pmatrix}
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\\
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&=
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\begin{pmatrix}
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u_yv_xw_y-u_yv_yw_x-u_zv_zw_x+u_zv_xw_z \\
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u_zv_yw_z-u_zv_zw_y-u_xv_xw_y+u_xv_yw_x \\
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u_xv_zw_x-u_xv_xw_z-u_yv_yw_z+u_yv_zw_y
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\end{pmatrix}
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\end{align}
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Ligne (1) est égale à ligne (2) donc $\vec{u}\land(\vec{v}\land\vec{w})=(\vec{u}\cdot\vec{w})\vec{v}-(\vec{u}\cdot\vec{v})\vec{w}$
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\end{document}
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