Formula_20605

\[\begin{gathered}(\boldsymbol{a}+\boldsymbol{b})_{i}=x_{i}+y_{i}, \quad(\lambda \boldsymbol{a})_{i}=\lambda \boldsymbol{a}_{i}, \\(\boldsymbol{a} \boldsymbol{b})=x_{1} y_{1}+x_{2} y_{2}+x_{3} y_{3}, \\{\left[\begin{array}{ll}\boldsymbol{a} & \boldsymbol{b}\end{array}\right]=\left|\begin{array}{lll}\boldsymbol{e}_{1} & \boldsymbol{e}_{2} & \boldsymbol{e}_{3} \\x_{1} & x_{2} & x_{3} \\y_{1} & y_{2} & y_{3}\end{array}\right|, \quad(\boldsymbol{a}[\boldsymbol{b} \boldsymbol{c}])=\left|\begin{array}{lll}x_{1} & x_{2} & x_{3} \\y_{1} & y_{2} & y_{3} \\z_{1} & z_{2} & z_{3}\end{array}\right| .}\end{gathered}\]