\vec a \times \vec b = \left| \begin{array}{ccc} 
\vec i & \vec j & \vec k\\
\alpha_1 & \alpha_2 & \alpha_3\\
\beta_1 & \beta_2 & \beta_3\\
\end{array}\right| =
\left|\begin{array}{cc}
\alpha_2 & \alpha_3\\
\beta_2 & \beta_3\\
\end{array}\right| \vec i
- \left|\begin{array}{cc}
\alpha_1 & \alpha_3\\
\beta_1 & \beta_3\\
\end{array}\right| \vec j
+ \left|\begin{array}{cc}
\alpha_1 & \alpha_2\\
\beta_1 & \beta_2\\
\end{array}\right| \vec k.