【解析几何笔记】9. 向量的内积运算

9. 向量的内积运算

定义：有向量$\alpha ,\beta$，$\alpha \cdot \beta =|\alpha ||\beta |\cdot \cos <\alpha ,\beta >$（结果是一个数）

• $\alpha \cdot \beta =0\iff \alpha \perp \beta$
• $\alpha \cdot \alpha =|\alpha {|}^2\Rightarrow \sqrt{\alpha \cdot \alpha }$
• $<\alpha ,\beta >=\arccos \frac{\alpha \cdot \beta }{|\alpha ||\beta |}$
• 【交换律】$\alpha \cdot \beta =\beta \cdot \alpha$
• $P_{\beta }\alpha =|\alpha |\cos <\alpha ,\beta >{\beta }_0=|\alpha |\cos <\alpha ,\beta >\frac{\beta }{|\beta |}\Rightarrow \alpha \cdot \beta =\frac{P_{\beta }\alpha }{{\beta }_0}|\beta |$
• 【定理1.3】内积具有双线性性
  （1）$\alpha \cdot (\beta +\gamma )=\alpha \cdot \beta +\alpha \cdot \gamma$
  $(\alpha +\gamma )\cdot \beta =\alpha \cdot \beta +\gamma \cdot \alpha$
  （2）$\alpha \cdot (\lambda \beta )=\lambda \alpha \beta =(\lambda \alpha )\beta$
  【证】（1）$(\alpha +\gamma )\cdot \beta =\frac{P_{\beta }(\alpha +\gamma )}{{\beta }_0}|\beta |=\frac{P_{\beta }\alpha +P_{\beta }\gamma }{{\beta }_0}|\beta |({\beta }_0与分母的向量都平行)=\frac{P_{\beta }\alpha }{{\beta }_0}+\frac{P_{\beta }\gamma }{{\beta }_0}=\alpha \cdot \beta +\alpha \cdot \gamma$
  （2）$(\lambda \alpha )\cdot \beta =\frac{P_{\beta }\lambda \alpha }{{\beta }_0}|\beta |=\frac{\lambda p_{\beta }\alpha }{{\beta }_0}|\beta |=\lambda \alpha \beta$

【例】用向量的内积来证明余弦定理。

【例1.12】

9.1 内积的坐标计算公式

有空间仿射坐标系$[O:e_1,e_2,e_3]$，$\alpha =(a_1,a_2,a_3),\beta =(b_1,b_2,b_3)$即$\alpha =a_1{e}_1+a_2{e}_2+a_3{e}_3,\beta =b_1{e}_1+b_2{e}_2+b_3{e}_3$
则$\alpha \cdot \beta =(a_1{e}_1+a_2{e}_2+a_3{e}_3)(b_1{e}_1+b_2{e}_2+b_3{e}_3)=a_1{b}_1{e}_1^2+a_1{b}_1{e}_1{e}_2+a_1{b}_3{e}_1{e}_3+a_2{b}_1{e}_1{e}_2+a_2{b}_2{e}_2^2+a_2{b}_3{e}_2{e}_3+a_3{b}_1{e}_1{e}_3+a_3{b}_2{e}_2{e}_3+a_3{b}_3{e}_3^2$
若$[O:e_1,e_2,e_3]$是直角坐标系，$e_i\perp e_j(i\ne j),|e_i|=1\iff e_i{e}_j=0(i\ne j),e_i^2=1$，则$\alpha \cdot \beta =(a_1{e}_1+a_2{e}_2+a_3{e}_3)(b_1{e}_1+b_2{e}_2+b_3{e}_3)=a_1{b}_1+a_2{b}_2+a_3{b}_3$
【例14】有正四面体