Notion of the cross product and important properties.
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Orthogonality |
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Definition 1. Let \(\mathbf{v}=\begin{pmatrix} v_1\\v_2\\v_3
\end{pmatrix}\) and \(\mathbf{w}=\begin{pmatrix} w_1\\w_2\\
w_3\end{pmatrix}\) be vectors in \(\mathbb{R}^3\). Then the cross
product of \(\mathbf{v}\) and
\(\mathbf{w}\) is defined as follows:
\(\mathbf{v}\times \mathbf{w} :=
\begin{pmatrix}
v_2w_3-v_3w_2\\
v_3w_1-v_1w_3\\
v_1w_2-v_2w_1
\end{pmatrix} .\)
Mnemonic
\(\rightsquigarrow\) write the
vectors next to each other and
\(\phantom{lalala}\) below them add the
first two components again
\(\rightsquigarrow\) delete the first
row
\(\rightsquigarrow\) determine the
entries of \(\mathbf{v}\times
\mathbf{w}\) using the drawn crosses: \[\mathbf{v}\times \mathbf{w} =
\begin{pmatrix}
{\color{blue}v_2w_3-v_3w_2}\\
{\color{red}v_3w_1-v_1w_3}\\
{\color{green!40!black}v_1w_2-v_2w_1}
\end{pmatrix}\]
Important properties:
The cross product is only defined in \(\mathbb{R}^3\)
The vector \(\mathbf{v}\times
\mathbf{w}\) is orthogonal to \(\mathbf{v}\) and \(\mathbf{w}\).
The parallelogramm whose sides are given by \(\mathbf{v}\) and \(\mathbf{w}\) has an area of size \(\| \mathbf{v} \times \mathbf{w} \|\) .
Example 2. \[\begin{pmatrix}3\\-2\\1\end{pmatrix} \times
\begin{pmatrix}0\\3\\-3\end{pmatrix} = \begin{pmatrix}(-2)\cdot (-3) -
3\cdot 1\\1\cdot 0 - (-3)\cdot 3\\3\cdot 3 - 0 \cdot (-2)\end{pmatrix} =
\begin{pmatrix}3\\9\\9\end{pmatrix}\]
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