Several useful formulas: (a) $\vec A \cdot \vec B = \vec B \cdot \vec A; \mm $ (b) $\vec A \times \vec B = - \vec B \times \vec A$;
(c) If $\vec B=(B_1, B_2, B_3)$ and $ \vec C =(C_1,C_2, C_3)$, then
$ \vec B\times \vec C = \left|\begin{array}{ccc} \vec i & \vec j &\vec k \\
B_1 & B_2 & B_3 \\
C_1 & C_2 & C_3 \end{array}\right| = \left|\begin{array}{cc}
B_2 & B_3 \\
C_2 & C_3 \end{array}\right| \vec i + \left|\begin{array}{cc}
B_3 & B_1 \\
C_3 & C_1 \end{array}\right| \vec j + \left|\begin{array}{cc}
B_1 & B_2 \\
C_1 & C_2 \end{array}\right| \vec k $