外積に外積を組み合わされた式
$$
\begin{eqnarray}
\left(\mathbf{A}\times\mathbf{B}\right)\times\mathbf{C}
&=&
\begin{vmatrix}
\mathbf{i}&\mathbf{j}&\mathbf{k}\\
x_a&y_a&z_a\\
x_b&y_b&z_b\\
\end{vmatrix}
\cdot
\left(x_c\mathbf{i}+y_c\mathbf{j}+z_c\mathbf{k}\right)
\\&=&
\left(
\left(y_az_b-y_bz_a\right)\mathbf{i}
+\left(x_bz_a-x_az_b\right)\mathbf{j}
+\left(x_ay_b-x_by_a\right)\mathbf{k}
\right)
\times
\left(
x_c\mathbf{i}
+y_c\mathbf{j}
+z_c\mathbf{k}
\right)
\\&=&
\begin{vmatrix}
\mathbf{i}&\mathbf{j}&\mathbf{k}\\
\left(y_az_b-y_bz_a\right)\mathbf{i}
&\left(x_bz_a-x_az_b\right)\mathbf{j}
&\left(x_ay_b-x_by_a\right)\mathbf{k}\\
x_c&y_c&z_c\\
\end{vmatrix}
\\&=&
\left\{ \left(x_bz_a-x_az_b\right)z_c-\left(x_ay_b-x_by_a\right)y_c \right\}\mathbf{i}
\\&&+\left\{ \left(x_ay_b-x_by_a\right)x_c-\left(y_az_b-y_bz_a\right)z_c \right\}\mathbf{j}
\\&&+\left\{ \left(y_az_b-y_bz_a\right)y_c-\left(x_bz_a-x_az_b\right)x_c \right\}\mathbf{k}
\\&=&
\left\{ \left(x_b\;z_az_c-x_a\;z_bz_c\right)-\left(x_a\;y_by_c-x_b\;y_ay_c\right) \right\}\mathbf{i}
\\&&+\left\{ \left(y_b\;x_ax_c-y_a\;x_bx_c\right)-\left(y_a\;z_bz_c-y_b\;z_az_c\right) \right\}\mathbf{j}
\\&&+\left\{ \left(z_b\;y_ay_c-z_a\;y_by_c\right)-\left(z_a\;x_bx_c-z_b\;x_ax_c\right) \right\}\mathbf{k}
\\&=&
\left\{\left(x_b\;z_az_c-x_a\;z_bz_c\right)-\left(x_a\;y_by_c-x_b\;y_ay_c\right) {\color{red}+x_b\;x_ax_c-x_a\;x_bx_c} \right\}\mathbf{i}
\\&&+\left\{\left(y_b\;x_ax_c-y_a\;x_bx_c\right)-\left(y_a\;z_bz_c-y_b\;z_az_c\right) {\color{red}+y_b\;y_ay_c-y_a\;y_by_c} \right\}\mathbf{j}
\\&&+\left\{\left(z_b\;y_ay_c-z_a\;y_by_c\right)-\left(z_a\;x_bx_c-z_b\;x_ax_c\right) {\color{red}+z_b\;z_az_c-z_a\;z_bz_c} \right\}\mathbf{k}
\\&=&
\left\{
\left(x_b\;x_ax_c+x_b\;y_ay_c+x_b\;z_az_c\right)\mathbf{i}
+\left(y_b\;x_ax_c+y_b\;y_ay_c+y_b\;z_az_c\right)\mathbf{j}
+\left(z_b\;x_ax_c+z_b\;y_ay_c+z_b\;z_az_c\right)\mathbf{k}
\right\}
\\&&-\left\{
\left(x_a\;x_bx_c+x_a\;y_by_c+x_a\;z_bz_c\right)\mathbf{i}
+\left(y_a\;x_bx_c+y_a\;y_by_c+y_a\;z_bz_c\right)\mathbf{j}
+\left(z_a\;x_bx_c+z_a\;y_by_c+z_a\;z_bz_c\right)\mathbf{k}
\right\}
\\&=&
\left\{
\left(x_ax_c+y_ay_c+z_az_c\right)x_b\;\mathbf{i}
+\left(x_ax_c+y_ay_c+z_az_c\right)y_b\;\mathbf{j}
+\left(x_ax_c+y_ay_c+z_az_c\right)z_b\;\mathbf{k}
\right\}
\\&&-\left\{
\left(x_bx_c+y_by_c+z_bz_c\right)x_a\;\mathbf{i}
+\left(x_bx_c+y_by_c+z_bz_c\right)y_a\;\mathbf{j}
+\left(x_bx_c+y_by_c+z_bz_c\right)z_a\;\mathbf{k}
\right\}
\\&=&
\left(x_ax_c+y_ay_c+z_az_c\right)\left(x_b\;\mathbf{i}+y_b\;\mathbf{j}+z_b\;\mathbf{k}\right)
-\left(x_bx_c+y_by_c+z_bz_c\right)\left(x_a\;\mathbf{i}+y_a\;\mathbf{j}+z_a\;\mathbf{k}\right)
\\&=&\left(\mathbf{A}\cdot\mathbf{C}\right)\mathbf{B}-\left(\mathbf{B}\cdot\mathbf{C}\right)\mathbf{A}
\end{eqnarray}
$$
$$
\begin{eqnarray}
\mathbf{A}\times\left(\mathbf{B}\times\mathbf{C}\right)
&=&
\left(x_a\mathbf{i}+y_a\mathbf{j}+z_a\mathbf{k}\right)
\times
\begin{vmatrix}
\mathbf{i}&\mathbf{j}&\mathbf{k}\\
x_b&y_b&z_b\\
x_c&y_c&z_c\\
\end{vmatrix}
\\&=&
\left(
x_a\mathbf{i}
+y_a\mathbf{j}
+z_a\mathbf{k}
\right)
\times
\left(
\left(y_bz_c-y_cz_b\right)\mathbf{i}
+\left(x_cz_b-x_bz_c\right)\mathbf{j}
+\left(x_by_c-x_cy_b\right)\mathbf{k}
\right)
\\&=&
\begin{vmatrix}
\mathbf{i}&\mathbf{j}&\mathbf{k}\\
x_a&y_a&z_a\\
\left(y_bz_c-y_cz_b\right)&\left(x_cz_b-x_bz_c\right)&\left(x_by_c-x_cy_b\right)\\
\end{vmatrix}
\\&=&
\left\{ y_a\left(x_by_c-x_cy_b\right)-z_a\left(x_cz_b-x_bz_c\right) \right\} \mathbf{i}
\\&&+ \left\{ z_a\left(y_bz_c-y_cz_b\right)-x_a\left(x_by_c-x_cy_b\right) \right\} \mathbf{j}
\\&&+ \left\{ x_a\left(x_cz_b-x_bz_c\right)-y_a\left(y_bz_c-y_cz_b\right) \right\} \mathbf{k}
\\&=&
\left\{ \left(x_b\;y_ay_c-x_c\;y_ay_b\right)-\left(x_c\;z_az_b-x_b\;z_az_c\right) \right\} \mathbf{i}
\\&&+ \left\{ \left(y_b\;z_az_c-y_c\;z_az_b\right)-\left(y_c\;x_ax_b-y_b\;x_ax_c\right) \right\} \mathbf{j}
\\&&+ \left\{ \left(z_b\;x_ax_c-z_c\;x_ax_b\right)-\left(z_c\;y_ay_b-z_b\;y_ay_c\right) \right\} \mathbf{k}
\\&=&
\left\{ \left(x_b\;y_ay_c-x_c\;y_ay_b\right)-\left(x_c\;z_az_b-x_b\;z_az_c\right) {\color{red}+x_b\;x_ax_c-x_c\;x_ax_b} \right\} \mathbf{i}
\\&&+ \left\{ \left(y_b\;z_az_c-y_c\;z_az_b\right)-\left(y_c\;x_ax_b-y_b\;x_ax_c\right) {\color{red}+y_b\;y_ay_c-y_c\;y_ay_b} \right\} \mathbf{j}
\\&&+ \left\{ \left(z_b\;x_ax_c-z_c\;x_ax_b\right)-\left(z_c\;y_ay_b-z_b\;y_ay_c\right) {\color{red}+z_b\;z_az_c-z_c\;z_az_b} \right\} \mathbf{k}
\\&=&
\left\{
\left( x_b\;x_ax_c+x_b\;y_ay_c+x_b\;z_az_c \right) \mathbf{i}
+ \left( y_b\;x_ax_c+y_b\;y_ay_c+y_b\;z_az_c \right) \mathbf{j}
+ \left( z_b\;x_ax_c+z_b\;y_ay_c+z_b\;z_az_c \right) \mathbf{k}
\right\}
\\&&-\left\{
\left( x_c\;x_ax_b+x_c\;y_ay_b+x_c\;z_az_b \right) \mathbf{i}
+ \left( y_c\;x_ax_b+y_c\;y_ay_b+y_c\;z_az_b \right) \mathbf{j}
+ \left( z_c\;x_ax_b+z_c\;y_ay_b+z_c\;z_az_b \right) \mathbf{k}
\right\}
\\&=&
\left\{
\left( x_ax_c+y_ay_c+z_az_c \right) x_b\;\mathbf{i}
+ \left( x_ax_c+y_ay_c+z_az_c \right) y_b\;\mathbf{j}
+ \left( x_ax_c+y_ay_c+z_az_c \right) z_b\;\mathbf{k}
\right\}
\\&&-\left\{
\left( x_ax_b+y_ay_b+z_az_b \right) x_c\;\mathbf{i}
+ \left( x_ax_b+y_ay_b+z_az_b \right) y_c\;\mathbf{j}
+ \left( x_ax_b+y_ay_b+z_az_b \right) z_c\;\mathbf{k}
\right\}
\\&=&
\left( x_ax_c+y_ay_c+z_az_c \right) \left(x_b\;\mathbf{i}+y_b\;\mathbf{j}+z_b\;\mathbf{k}\right)
- \left( x_ax_b+y_ay_b+z_az_b \right) \left(x_c\;\mathbf{i}+y_c\;\mathbf{j}+z_c\;\mathbf{k}\right)
\\&=&\left(\mathbf{A}\cdot\mathbf{C}\right)\mathbf{B}-\left(\mathbf{A}\cdot\mathbf{B}\right)\mathbf{C}
\\
\\
\left(\mathbf{A}\times\mathbf{B}\right)\times\mathbf{C}
&=&-\mathbf{C}\times\left(\mathbf{A}\times\mathbf{B}\right)
\;\cdots\;\mathbf{A}\times\mathbf{B}=-\mathbf{B}\times\mathbf{A}
\\&=&\left((-\mathbf{C})\cdot\mathbf{B}\right)\mathbf{A}-\left((-\mathbf{C})\cdot\mathbf{A}\right)\mathbf{B}
\;\cdots\;\mathbf{A}\times\left(\mathbf{B}\times\mathbf{C}\right)=\left(\mathbf{A}\cdot\mathbf{C}\right)\mathbf{B}-\left(\mathbf{A}\cdot\mathbf{B}\right)\mathbf{C}
\\&=&\left(\mathbf{A}\cdot\mathbf{C}\right)\mathbf{B}-\left(\mathbf{B}\cdot\mathbf{C}\right)\mathbf{A}
\;\cdots\;\mathbf{A}\cdot\mathbf{B}=\mathbf{B}\cdot\mathbf{A},\;\left(\left(-\mathbf{A}\right)\cdot\mathbf{B}\right)=-\left(\mathbf{A}\cdot\mathbf{B}\right)
\end{eqnarray}
$$
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