Vektorja zapiši kot linearno kombinacijo vektorjev $\overset{\rightharpoonup}{i},\overset{\rightharpoonup}{j},\overset{\rightharpoonup}{k}$ in uporabi lastnosti vektoskega produkta.

$$\overset{\rightharpoonup}{a}\times\overset{\rightharpoonup}{b}=(a_1\overset{\rightharpoonup}{i}+a_2\overset{\rightharpoonup}{j}+a_3\overset{\rightharpoonup}{k})(b_1\overset{\rightharpoonup}{i}+b_2\overset{\rightharpoonup}{j}+b_3\overset{\rightharpoonup}{k})=$$ $$=a_1b_1\overset{\rightharpoonup}{i}\times\overset{\rightharpoonup}{i}+a_1b_2\overset{\rightharpoonup}{i}\times\overset{\rightharpoonup}{j}+a_1b_3\overset{\rightharpoonup}{i}\times\overset{\rightharpoonup}{k}+$$ $$+a_2b_1\overset{\rightharpoonup}{j}\times\overset{\rightharpoonup}{i}+a_2b_2\overset{\rightharpoonup}{j}\times\overset{\rightharpoonup}{j}+a_2b_3\overset{\rightharpoonup}{j}\times\overset{\rightharpoonup}{k}+$$ $$+a_3b_1\overset{\rightharpoonup}{k}\times\overset{\rightharpoonup}{i}+a_3b_2\overset{\rightharpoonup}{k}\times\overset{\rightharpoonup}{j}+a_3b_3\overset{\rightharpoonup}{k}\times\overset{\rightharpoonup}{k}=$$ $$=a_1b_1\overset{\rightharpoonup}{0}+a_1b_2\overset{\rightharpoonup}{k}-a_1b_3\overset{\rightharpoonup}{j}-a_2b_1\overset{\rightharpoonup}{k}+$$ $$+a_2b_2\overset{\rightharpoonup}{0}+a_2b_3\overset{\rightharpoonup}{i}+a_3b_1\overset{\rightharpoonup}{j}-a_3b_2\overset{\rightharpoonup}{i}+a_3b_3\overset{\rightharpoonup}{0}=$$ $$=(a_2b_3-a_3b_2)\overset{\rightharpoonup}{i}+(a_3b_1-a_1b_3)\overset{\rightharpoonup}{j}+(a_1b_2-a_2b_1)\overset{\rightharpoonup}{k}=$$ $$=(a_2b_3-a_3b_2,a_3b_1-a_1b_3,a_1b_2-a_2b_1)$$

Vrednost determinante je enaka vektorskemu produktu vektorjev $\overset{\rightharpoonup}{a}=(a_1,a_2,a_3)$ in $\overset{\rightharpoonup}{b}=(b_1,b_2,b_3)$.