td/mecanique_du_point_td_1.tex

\documentclass{article}

\usepackage{mathtools}
\usepackage{amssymb}

\title{Mécanique du point - TD 1 \\ NON CORRIGÉ}
\author{Timéo Pochin}

\begin{document}
    \maketitle

    \section*{Exercice 1}

    \subsection*{a)}
    \begin{align}
    \vec{u}\land\vec{v}
    \nonumber
    &=
    \begin{pmatrix}
        1 \\
        2 \\
        3
    \end{pmatrix}
    \land
    \begin{pmatrix}
        4 \\
        5 \\
        6
    \end{pmatrix}
    \\ \nonumber
    &=
    \begin{pmatrix}
        2\cdot 6-5\cdot 3 \\
        4\cdot 3-1\cdot 6 \\
        1\cdot 5-4\cdot 2
    \end{pmatrix}
    \\ \nonumber
    &=
    \begin{pmatrix}
        -3 \\
        6 \\
        -3
    \end{pmatrix}
    \\ \nonumber
    &=-3\vec{e_x}+6\vec{e_y}-3\vec{e_z}
    \end{align}

    \subsection*{b)}
    \begin{align}
    \vec{v}\land\vec{u}
    \nonumber
    &=
    \begin{pmatrix}
        4 \\
        5 \\
        6
    \end{pmatrix}
    \land
    \begin{pmatrix}
        1 \\
        2 \\
        3
    \end{pmatrix}
    \\ \nonumber
    &=
    \begin{pmatrix}
        5\cdot 3-2\cdot 6 \\
        1\cdot 6-4\cdot 3 \\
        4\cdot 2-1\cdot 5
    \end{pmatrix}
    \\ \nonumber
    &=
    \begin{pmatrix}
        3 \\
        -6 \\
        3
    \end{pmatrix}
    \\ \nonumber
    &=3\vec{e_x}-6\vec{e_y}+3\vec{e_z}
    \end{align}

    \subsection*{c)}
    \begin{align}
    \vec{u}\cdot\vec{v}
    \nonumber
    &=
    \begin{pmatrix}
        1 \\
        2 \\
        3
    \end{pmatrix}
    \cdot
    \begin{pmatrix}
        4 \\
        5 \\
        6
    \end{pmatrix}
    \\ \nonumber
    &=1\cdot 4+2\cdot 5+3\cdot 6\\ \nonumber
    &=32
    \end{align}

    \subsection*{d)}
    \begin{align}
    \vec{u}\cdot(\vec{u}\land\vec{v})
    \nonumber
    &=
    \begin{pmatrix}
        1 \\
        2 \\
        3
    \end{pmatrix}
    \cdot
    \begin{pmatrix}
        -3 \\
        6 \\
        -3
    \end{pmatrix}
    \\ \nonumber
    &=1\cdot-3+2\cdot 6+3\cdot -3\\ \nonumber
    &=0
    \end{align}

    \subsection*{e)}
    \begin{align}
    \|\vec{u}\|
    \nonumber
    &=\sqrt{1^2+2^2+3^2}
    \\ \nonumber
    &=\sqrt{14}
    \end{align}

    \subsection*{f)}
    \begin{align}
    \|\vec{u}+\vec{v}\|
    \nonumber
    &=
    \left\|
    \begin{pmatrix}
        1 \\
        2 \\
        3
    \end{pmatrix}
    +
    \begin{pmatrix}
        4 \\
        5 \\
        6
    \end{pmatrix}
    \right\|
    \\ \nonumber
    &=
    \left\|
    \begin{pmatrix}
        5 \\
        7 \\
        9
    \end{pmatrix}
    \right\|
    \\ \nonumber
    &=\sqrt{5^2+7^2+9^2}
    \\ \nonumber
    &=\sqrt{155}
    \end{align}

    \subsection*{g)}
    \begin{align}
    (\vec{u}\land\vec{v})\land\vec{w}
    \nonumber
    &=
    \begin{pmatrix}
        -3 \\
        6 \\
        -3
    \end{pmatrix}
    \land
    \begin{pmatrix}
        7 \\
        8 \\
        9
    \end{pmatrix}
    \\ \nonumber
    &=
    \begin{pmatrix}
        6\cdot 9-8\cdot-3 \\
        7\cdot-3-9\cdot-3 \\
        -3\cdot 8-7\cdot 6
    \end{pmatrix}
    \\ \nonumber
    &=
    \begin{pmatrix}
        78 \\
        6 \\
        -66
    \end{pmatrix}
    \\ \nonumber
    &=78\vec{e_x}+6\vec{e_y}-66\vec{e_z}
    \end{align}

    \subsection*{h)}
    \begin{align}
    \vec{u}\land(\vec{v}\land\vec{w})
    \nonumber
    &=
    \begin{pmatrix}
        1 \\
        2 \\
        3
    \end{pmatrix}
    \land
    \left(
    \begin{pmatrix}
        4 \\
        5 \\
        6
    \end{pmatrix}
    \land
    \begin{pmatrix}
        7 \\
        8 \\
        9
    \end{pmatrix}
    \right)
    \\ \nonumber
    &=
    \begin{pmatrix}
        1 \\
        2 \\
        3
    \end{pmatrix}
    \land
    \begin{pmatrix}
        5\cdot 9-8\cdot 6 \\
        7\cdot 6-4\cdot 9 \\
        4\cdot 8-7\cdot 5
    \end{pmatrix}
    \\ \nonumber
    &=
    \begin{pmatrix}
        1 \\
        2 \\
        3
    \end{pmatrix}
    \land
    \begin{pmatrix}
        -3 \\
        6 \\
        -3
    \end{pmatrix}
    \\ \nonumber
    &=
    \begin{pmatrix}
        2\cdot-3-6\cdot 3 \\
        -3\cdot 3-1\cdot-3 \\
        1\cdot 6--3\cdot 3
    \end{pmatrix}
    \\ \nonumber
    &=
    \begin{pmatrix}
        -24 \\
        -6 \\
        12
    \end{pmatrix}
    \\ \nonumber
    &= -24\vec{e_x}-6\vec{e_y}+12\vec{e_z}
    \end{align}

\end{document}