We have u × v = ∣ u 1 u 2 u 3 v 1 v 2 v 3 e 1 e 2 e 3 ∣ = u\times v=\begin{vmatrix}
u_1&u_2&u_3\\
v_1&v_2&v_3\\
e_1&e_2&e_3
\end{vmatrix}= u × v = ∣ ∣ u 1 v 1 e 1 u 2 v 2 e 2 u 3 v 3 e 3 ∣ ∣ =
= ( ∣ u 2 u 3 v 2 v 3 ∣ , − ∣ u 1 u 3 v 1 v 3 ∣ , ∣ u 1 u 2 v 1 v 2 ∣ ) =\left(\begin{vmatrix}
u_2&u_3\\
v_2&v_3\\
\end{vmatrix},-\begin{vmatrix}
u_1&u_3\\
v_1&v_3\\
\end{vmatrix},\begin{vmatrix}
u_1&u_2\\
v_1&v_2\\
\end{vmatrix}\right) = ( ∣ ∣ u 2 v 2 u 3 v 3 ∣ ∣ , − ∣ ∣ u 1 v 1 u 3 v 3 ∣ ∣ , ∣ ∣ u 1 v 1 u 2 v 2 ∣ ∣ )
Then ( u , u × v ) = u 1 ∣ u 2 u 3 v 2 v 3 ∣ − u 2 ∣ u 1 u 3 v 1 v 3 ∣ + u 3 ∣ u 1 u 2 v 1 v 2 ∣ = (u,u\times v)=u_1\begin{vmatrix}
u_2&u_3\\
v_2&v_3\\
\end{vmatrix}-u_2\begin{vmatrix}
u_1&u_3\\
v_1&v_3\\
\end{vmatrix}+u_3\begin{vmatrix}
u_1&u_2\\
v_1&v_2\\
\end{vmatrix}= ( u , u × v ) = u 1 ∣ ∣ u 2 v 2 u 3 v 3 ∣ ∣ − u 2 ∣ ∣ u 1 v 1 u 3 v 3 ∣ ∣ + u 3 ∣ ∣ u 1 v 1 u 2 v 2 ∣ ∣ =
= ∣ u 1 u 2 u 3 v 1 v 2 v 3 u 1 u 2 u 3 ∣ = 0 =\begin{vmatrix}
u_1&u_2&u_3\\
v_1&v_2&v_3\\
u_1&u_2&u_3
\end{vmatrix}=0 = ∣ ∣ u 1 v 1 u 1 u 2 v 2 u 2 u 3 v 3 u 3 ∣ ∣ = 0 and ( v , u × v ) = v 1 ∣ u 2 u 3 v 2 v 3 ∣ − v 2 ∣ u 1 u 3 v 1 v 3 ∣ + v 3 ∣ u 1 u 2 v 1 v 2 ∣ = (v,u\times v)=v_1\begin{vmatrix}
u_2&u_3\\
v_2&v_3\\
\end{vmatrix}-v_2\begin{vmatrix}
u_1&u_3\\
v_1&v_3\\
\end{vmatrix}+v_3\begin{vmatrix}
u_1&u_2\\
v_1&v_2\\
\end{vmatrix}= ( v , u × v ) = v 1 ∣ ∣ u 2 v 2 u 3 v 3 ∣ ∣ − v 2 ∣ ∣ u 1 v 1 u 3 v 3 ∣ ∣ + v 3 ∣ ∣ u 1 v 1 u 2 v 2 ∣ ∣ =
= ∣ u 1 u 2 u 3 v 1 v 2 v 3 v 1 v 2 v 3 ∣ = 0 =\begin{vmatrix}
u_1&u_2&u_3\\
v_1&v_2&v_3\\
v_1&v_2&v_3
\end{vmatrix}=0 = ∣ ∣ u 1 v 1 v 1 u 2 v 2 v 2 u 3 v 3 v 3 ∣ ∣ = 0
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