Abstellraum: Strukturwissenschaften: Vektoren

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[Bearbeiten] Euklidischer Raum

\mathbf{v}\in\mathbb{R}^n\Longleftrightarrow\mathbf{v}=\begin{pmatrix} \mathit{v}_0 \\ \mathit{v}_1 \\ \vdots  \\ \mathit{v}_{n-1} \end{pmatrix}\rm{mit} \mathit{v_i}\in\mathbb{R}, \mathit{i} = 0,\cdots,n-1

[Bearbeiten] Addition

\mathbf{u} + \mathbf{v} = \begin{pmatrix} \mathit{u}_0 \\ \mathit{u}_1 \\ \vdots  \\ \mathit{u}_{n-1} \end{pmatrix} +
\begin{pmatrix} \mathit{v}_0 \\ \mathit{v}_1 \\ \vdots  \\ \mathit{v}_{n-1} \end{pmatrix} = \begin{pmatrix} \mathit{u}_0 + \mathit{v}_0 \\ \mathit{u}_1 + \mathit{v}_1\\ \vdots  \\ \mathit{u}_{n-1} + \mathit{v}_{n-1} \end{pmatrix}\in\mathbb{R}^n

[Bearbeiten] Subtraktion

\mathbf{u} - \mathbf{v} = \begin{pmatrix} \mathit{u}_0 \\ \mathit{u}_1 \\ \vdots  \\ \mathit{u}_{n-1} \end{pmatrix} -
\begin{pmatrix} \mathit{v}_0 \\ \mathit{v}_1 \\ \vdots  \\ \mathit{v}_{n-1} \end{pmatrix} = \begin{pmatrix} \mathit{u}_0 - \mathit{v}_0 \\ \mathit{u}_1 - \mathit{v}_1\\ \vdots  \\ \mathit{u}_{n-1} - \mathit{v}_{n-1} \end{pmatrix}\in\mathbb{R}^n

[Bearbeiten] Multiplikation mit einem Skalar

\mathit{a}\mathbf{u} = \begin{pmatrix} \mathit{au}_0 \\ \mathit{au}_1 \\ \vdots  \\ \mathit{au}_{n-1} \end{pmatrix}\in\mathbb{R}^n

[Bearbeiten] Kreuzprodukt

\mathbf{w}=\begin{pmatrix} \mathit{w}_x \\ \mathit{w}_y \\ \mathit{w}_{z} \end{pmatrix} 
= \mathbf{u} \times \mathbf{v} = \begin{pmatrix} \mathit{u_yv_z}-\mathit{u_zv_y} \\ \mathit{u_zv_x}-\mathit{u_xv_z}  \\ \mathit{u_xv_y}-\mathit{u_yv_x} \end{pmatrix}

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