Is the fact that

[tex]\vec{a}\times\vec{b} = \det \begin{bmatrix}\hat{e}_1& \hat{e}_2 & \hat{e}_3 \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{bmatrix}[/tex]

just a coincedence (ie. a mneminic device) or does it have some deep mathematical significance?

edit: Also, is the cross product, like the dot product, defined for higher dimentions?

[tex]\vec{a}\times\vec{b} = \det \begin{bmatrix}\hat{e}_1& \hat{e}_2 & \hat{e}_3 \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{bmatrix}[/tex]

just a coincedence (ie. a mneminic device) or does it have some deep mathematical significance?

edit: Also, is the cross product, like the dot product, defined for higher dimentions?

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