Cross Product

Let’s define vector \vec{u} = (u_1, u_2, u_3) and \vec{v} = (v_1, v_2, v_3) and their cross product \vec{w}.
As known that \vec{w} = \vec{u} \times \vec{v} = (u_2v_3 - u_3v_2,  u_3v_1 - u_1v_3, u_1v_2 - u_2v_1), so we have:

    \[\vec{w} = \vec{u} \times \vec{v} = \begin{pmatrix} u_1 & u_2 & u_3 \end{pmatrix} \begin{pmatrix} 0 & -v_3 & v_2 \\ v_3 & 0 & -v_1 \\ -v_2 & v_1 & 0 \end{pmatrix}\]

The cross product is antisymmetric due to its definition.

    \[\vec{u} \times \vec{v} = -(\vec{v} \times \vec{u})\]

Dot Product

    \[\vec{u} \cdot \vec{v} = ||\vec{u}|| \,||\vec{v}||cos\theta\]

So we can write the dot product of two vectors as the matrix form.

    \[\vec{u} \cdot \vec{v} =  \begin{pmatrix} u_1 & u_2 & u_3 \end{pmatrix} \begin{pmatrix} v_1 \\  v_2 \\  v_3 \end{pmatrix}\]

Tensor Product

If we have the vector T:

    \[\vec{t} = (\vec{u} \cdot \vec{v})\vec{w}\]

We can write it by matrix based on the previous knowledges.

    \[\vec{t} =  \begin{pmatrix} u_1 & u_2 & u_3 \end{pmatrix} \begin{pmatrix} v_1 \\  v_2 \\  v_3 \end{pmatrix} \begin{pmatrix} w_1 & w_2 & w_3 \end{pmatrix} = \begin{pmatrix} u_1 & u_2 & u_3 \end{pmatrix} \begin{pmatrix} v_1w_1 & v_1w_2 & v_1w_3 \\ v_2w_1 & v_2w_2 & v_2w_3 \\ v_3w_1 & v_3w_2 & v_3w_3 \end{pmatrix}\]

    \[\vec{t} = \vec{u}(\vec{v} \otimes \vec{w})\]

\vec{v} \otimes \vec{w} is called tensor product of \vec{v} and \vec{w}.

Categories: Math

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