Cross Product (Dimension 3)

$$[a_1{a}_2{a}_3]\times [b_1{b}_2{b}_3]=[a_2\ast b_3-a_3\ast b_2{a}_3\ast b_1-b_3\ast a_1{a}_1\ast b_2-a_2\ast b_1]$$

Dot Product

$$[a_1{a}_2{a}_3]\cdot [a_2\ast b_3-a_3\ast b_2{a}_3\ast b_1-b_3\ast a_1{a}_1\ast b_2-a_2\ast b_1]$$ $$\begin{gathered} [a_1{a}_2{a}_3]\cdot \left[\begin{array}{c}{a}_2\ast b_3-a_3\ast b_2\\ a_3\ast b_1-b_3\ast a_1\\ a_1\ast b_2-a_2\ast b_1\end{array}\right] \\ =a_1\ast (a_2\ast b_3-a_3\ast b_2)+a_2\ast (a_3\ast b_1-b_3\ast a_1)+a_3\ast (a_1\ast b_2-a_2\ast b_1) \\ =a_1\ast a_2\ast b_3-a_1\ast a_3\ast b_2+a_2\ast a_3\ast b_1-a_2\ast b_3\ast a_1+a_3\ast a_1\ast b_2-a_3\ast a_2\ast b_1 \\ =a_1\ast a_2\ast b_3-a_2\ast b_3\ast a_1+a_2\ast a_3\ast b_1-a_3\ast a_2\ast b_1+a_3\ast a_1\ast b_2-a_1\ast a_3\ast b_2 \\ =0+0+0 \\ =0 \end{gathered}$$