Cross Product, Dot Product, Tensor Product In Matrices Forms

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.

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