91影视

The vectors $\mathrm{A}$, B and C.

The triple scalar product for three vectors is written as $\mathbf{A}$.$\left(B脳C\right)$

The determinant is as follows.

$\mathbf{A}\mathbf{.}\left(B脳C\right)\mathbf{=}\left|\begin{array}{ccc}{A}_x& A_y& A_z\\ B_x& B_y& B_z\\ C_x& C_y& C_z\end{array}\right|$

The triple scalar products involving A, B and C is given as follows.

$$\begin{gathered} \mathrm{A}.\left(\mathrm{B}脳\mathrm{C}\right)={\mathrm{A}}_{\mathrm{x}}.{\left(\mathrm{B}脳\mathrm{C}\right)}_{\mathrm{x}}+{\mathrm{A}}_{\mathrm{y}}.{\left(\mathrm{B}脳\mathrm{C}\right)}_{\mathrm{y}}+{\mathrm{A}}_{\mathrm{z}}.{\left(\mathrm{B}脳\mathrm{C}\right)}_{\mathrm{z}}\mathrm{A}脳\left(\mathrm{B}脳\mathrm{C}\right) \\ =\left|\begin{array}{ccc}{\mathrm{A}}_{\mathrm{x}}& {\mathrm{A}}_{\mathrm{y}}& {\mathrm{A}}_{\mathrm{z}}\\ {\mathrm{B}}_{\mathrm{x}}& {\mathrm{B}}_{\mathrm{y}}& {\mathrm{B}}_{\mathrm{z}}\\ {\mathrm{C}}_{\mathrm{x}}& {\mathrm{C}}_{\mathrm{y}}& {\mathrm{C}}_{\mathrm{z}}\end{array}\right| \end{gathered}$$

On interchanging the rows of determinants, there will be a minus sign.

Verification of the below determinant is s follows.

$$\begin{gathered} \left(\mathrm{A}脳\mathrm{B}\right).\mathrm{C}=\mathrm{A}.\left(\mathrm{B}脳\mathrm{C}\right) \\ =\mathrm{C}.\left(\mathrm{A}脳\mathrm{B}\right) \\ =鈥�\left(\mathrm{A}脳\mathrm{C}\right).\mathrm{B} \end{gathered}$$

The determinant equivalent to$\mathrm{A}.\left(\mathrm{B}脳\mathrm{C}\right)$is obtained when two rows are interchanged.

$\mathrm{A}.\left(\mathrm{B}脳\mathrm{C}\right)=\left(\mathrm{B}脳\mathrm{C}\right).\mathrm{A}=\left(\mathrm{A}脳\mathrm{B}\right).\mathrm{C}=\mathrm{C}.\left(\mathrm{A}脳\mathrm{B}\right)=\left(\mathrm{C}脳\mathrm{A}\right).\mathrm{B}=\mathrm{B}.\left(\mathrm{C}脳\mathrm{A}\right)$

The determinant equivalent to$\left(鈥�\mathrm{A}.\left(\mathrm{B}脳\mathrm{C}\right)\right)$is obtained when one row is interchanged.

$\mathrm{B}.\left(\mathrm{A}脳\mathrm{C}\right)=\left(\mathrm{A}脳\mathrm{C}\right).\mathrm{B}=\left(\mathrm{B}脳\mathrm{A}\right).\mathrm{C}=\left(\mathrm{C}脳\mathrm{B}\right).\mathrm{A}=\mathrm{A}.\left(\mathrm{C}脳\mathrm{B}\right)=\mathrm{C}.\left(\mathrm{B}脳\mathrm{A}\right)=鈥�\mathrm{A}.\left(\mathrm{B}脳\mathrm{C}\right)$

Hence, the twelve triple scalar products involving A B and C are as follows.

$\mathrm{A}.\left(\mathrm{B}脳\mathrm{C}\right)=\left(\mathrm{B}脳\mathrm{C}\right).\mathrm{A}=\left(\mathrm{A}脳\mathrm{B}\right).\mathrm{C}=\mathrm{C}.\left(\mathrm{A}脳\mathrm{B}\right)=\left(\mathrm{C}脳\mathrm{A}\right).\mathrm{B}=\mathrm{B}.\left(\mathrm{C}脳\mathrm{A}\right)$