Cross Product

I have to apply it in my RPCN course.

For any vector v $\in$ $R^3$, $v=\left(\begin{array}{c}{v}_x\\ v_y\\ v_z\end{array}\right)$, the skew-symmetric matrix is always:

$$[\mathbf{v}{]}_{\mathbf{x}}=\left[\begin{array}{ccc}0& -v_z& v_y\\ v_z& 0& -v_x\\ -v_y& v_x& 0\end{array}\right]$$

Key Properties:

1. Diagonal is always zero
2. Antisymmetric: $[\mathbf{v}{]}_{\mathbf{x}}^T=-[\mathbf{v}{]}_{\mathbf{x}}$

So when we want to compute the cross product between two vectors $\mathbf{v}$ and $\mathbf{w}$, we do

$$\mathbf{v}\times \mathbf{w}=[\mathbf{v}{]}_{\mathbf{x}}\mathbf{w}=\left[\begin{array}{ccc}0& -v_z& v_y\\ v_z& 0& -v_x\\ -v_y& v_x& 0\end{array}\right]\left[\begin{array}{c}{w}_x\\ w_y\\ w_z\end{array}\right]$$

Or, if you think you can memorize

$$\left[\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}\right]\times \left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]=\left[\begin{array}{c}{a}_2{b}_3-a_3{b}_2\\ a_3{b}_1-a_1{b}_3\\ a_1{b}_2-a_2{b}_1\end{array}\right]$$