Calcul vectorial/Produs vectorial

Fie $\mathbf {a} =a_{1}\mathbf {i} +a_{2}\mathbf {j} +a_{3}\mathbf {k} ,\;\mathbf {b} =b_{1}\mathbf {i} +b_{2}\mathbf {j} +b_{3}\mathbf {k}$ vectori din $\mathbb {R} ^{3}.$

Produsul vectorial dintre $\mathbf {a} ,\mathbf {b} ,$ notat $\mathbf {a} \times \mathbf {b} ,$ este vectorul:

$\mathbf {a} \times \mathbf {b} ={\begin{vmatrix}a_{2}&a_{3}\\b_{2}&b_{3}\end{vmatrix}}\mathbf {i} -{\begin{vmatrix}a_{1}&a_{3}\\b_{1}&b_{3}\end{vmatrix}}\mathbf {j} +{\begin{vmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{vmatrix}}\mathbf {k} ,$

sau simbolic:

$\mathbf {a} \times \mathbf {b} ={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\end{vmatrix}}$.