问题
若向量 $\vec{a}$ $=$ $(x_{1}, y_{1}, z_{1})$, $\vec{b}$ $=$ $(x_{2}, y_{2}, z_{2})$, $\vec{c}$ $=$ $(x_{3}, y_{3}, z_{3})$, 向量 $\mathbf{\vec{i}}$, $\mathbf{\vec{j}}$, $\mathbf{\vec{k}}$ 分别是 $x$ 轴、$y$ 轴和 $z$ 轴上的单位向量，则向量积 $\vec{a}$ $\times$ $\vec{b}$ $=$ $?$选项
[A]. $\vec{a}$ $\times$ $\vec{b}$ $=$ $\begin{vmatrix} x_{1} & z_{1} \\ x_{2} & z_{2} \end{vmatrix} \mathbf{i}$ $-$ $\begin{vmatrix} y_{1} & z_{1} \\ y_{2} & z_{2} \end{vmatrix} \mathbf{j}$ $+$ $\begin{vmatrix} x_{1} & y_{1} \\ x_{2} & y_{2} \end{vmatrix} \mathbf{k}$[B]. $\vec{a}$ $\times$ $\vec{b}$ $=$ $\begin{vmatrix} y_{1} & z_{1} \\ y_{2} & z_{2} \end{vmatrix} \mathbf{i}$ $+$ $\begin{vmatrix} x_{1} & z_{1} \\ x_{2} & z_{2} \end{vmatrix} \mathbf{j}$ $-$ $\begin{vmatrix} x_{1} & y_{1} \\ x_{2} & y_{2} \end{vmatrix} \mathbf{k}$
[C]. $\vec{a}$ $\times$ $\vec{b}$ $=$ $\begin{vmatrix} y_{1} & z_{1} \\ y_{2} & z_{2} \end{vmatrix} \mathbf{i}$ $+$ $\begin{vmatrix} x_{1} & z_{1} \\ x_{2} & z_{2} \end{vmatrix} \mathbf{j}$ $+$ $\begin{vmatrix} x_{1} & y_{1} \\ x_{2} & y_{2} \end{vmatrix} \mathbf{k}$
[D]. $\vec{a}$ $\times$ $\vec{b}$ $=$ $\begin{vmatrix} y_{1} & z_{1} \\ y_{2} & z_{2} \end{vmatrix} \mathbf{i}$ $-$ $\begin{vmatrix} x_{1} & z_{1} \\ x_{2} & z_{2} \end{vmatrix} \mathbf{j}$ $+$ $\begin{vmatrix} x_{1} & y_{1} \\ x_{2} & y_{2} \end{vmatrix} \mathbf{k}$
$\textcolor{red}{\vec{a}}$ $\textcolor{green}{\times}$ $\textcolor{red}{\vec{b}}$ $=$
$\begin{vmatrix}\mathbf{\textcolor{orange}{i}} & \mathbf{\textcolor{orange}{j}} & \mathbf{\textcolor{orange}{k}} \\ x_{\textcolor{cyan}{1}} & y_{\textcolor{cyan}{1}} & z_{\textcolor{cyan}{1}} \\ x_{\textcolor{blue}{2}} & y_{\textcolor{blue}{2}} & z_{\textcolor{blue}{2}} \end{vmatrix}$ $=$
$\begin{vmatrix} y_{\textcolor{cyan}{1}} & z_{\textcolor{cyan}{1}} \\ y_{\textcolor{blue}{2}} & z_{\textcolor{blue}{2}} \end{vmatrix} \mathbf{\textcolor{orange}{i}}$ $\textcolor{green}{-}$ $\begin{vmatrix} x_{\textcolor{cyan}{1}} & z_{\textcolor{cyan}{1}} \\ x_{\textcolor{blue}{2}} & z_{\textcolor{blue}{2}} \end{vmatrix} \mathbf{\textcolor{orange}{j}}$ $\textcolor{green}{+}$ $\begin{vmatrix} x_{\textcolor{cyan}{1}} & y_{\textcolor{cyan}{1}} \\ x_{\textcolor{blue}{2}} & y_{\textcolor{blue}{2}} \end{vmatrix} \mathbf{\textcolor{orange}{k}}$