混合积 vs 行列式

下面在空间直角坐标系中建立混合积的坐标表达式.

设

• $\mathbf{a}=a_x\mathbf{i}+a_y\mathbf{j}+a_z\mathbf{k}$,
• $\mathbf{b}=b_x\mathbf{i}+b_y\mathbf{j}+b_z\mathbf{k}$,
• $\mathbf{c}=c_x\mathbf{i}+c_y\mathbf{j}+c_z\mathbf{k}$

由

$$\mathbf{a}\times \mathbf{b}=\left|\begin{array}{ll}{a}_y& a_z\\ b_y& b_z\end{array}\right|\mathbf{i}+\left|\begin{array}{ll}{a}_z& a_x\\ b_z& b_x\end{array}\right|\mathbf{j}+\left|\begin{array}{ll}{a}_x& a_y\\ b_x& b_y\end{array}\right|\mathbf{k}$$

可得

$$\begin{array}{rl}\left(\mathbf{a},\mathbf{b},\mathbf{c}\right)& =\left|\begin{array}{ll}{a}_y& a_z\\ b_y& b_z\end{array}\right|c_x+\left|\begin{array}{ll}{a}_z& a_x\\ b_z& b_x\end{array}\right|c_y+\left|\begin{array}{ll}{a}_x& a_y\\ b_x& b_y\end{array}\right|c_z\\ & =\left|\begin{array}{lll}{a}_x& a_y& a_z\\ b_x& b_y& b_z\\ c_x& c_y& c_z\end{array}\right|.\end{array}$$

从而有

$$\left(\mathbf{a},\mathbf{b},\mathbf{c}\right)=\left|\begin{array}{lll}{a}_x& b_x& c_x\\ a_y& b_y& c_y\\ a_z& b_z& c_z\end{array}\right|.$$