版块 13 向量代数

A. 实用解题工具

第 1 组: 向量数量积与向量积

1. 数量积(点积,内积) :

$$\{\begin{array}{l}\overrightarrow{a}\cdot \overrightarrow{b}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta \left(\theta \text{ 是向量间夹角 }\right)\\ \text{ 坐标形式 : }\{\begin{array}{l}\overrightarrow{a}=\left\{x_1,y_1,z_1\right\},\overrightarrow{b}=\left\{x_2,y_2,z_2\right\}\\ \overrightarrow{a}\cdot \overrightarrow{b}=x_1{x}_2+y_1{y}_2+z_1{z}_2.\end{array}\end{array}$$

2. 向量积(叉乘,外积):

$$\{\begin{array}{l}\text{ 表达式 : }\overrightarrow{c}=\overrightarrow{a}\times \overrightarrow{b}\\ \text{ 大小 : }\left|\overrightarrow{c}\right|=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\sin \theta \text{ (}\theta \text{ 是向量间夹角) }\\ \text{ 公式 : }\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{lll}\overrightarrow{i}& \overrightarrow{j}& \overrightarrow{k}\\ x_1& y_1& z_1\\ x_2& y_2& z_2\end{array}\right|=\left|\begin{array}{ll}{y}_1& z_1\\ y_2& z_2\end{array}\right|\overrightarrow{i}-\left|\begin{array}{ll}{x}_1& z_1\\ x_2& z_2\end{array}\right|\overrightarrow{j}+\left|\begin{array}{ll}{x}_1& y_1\\ x_2& y_2\end{array}\right|\overrightarrow{k}.\end{array}$$

第 2 组: 向量混合积

1. 混合积:

$$\{\begin{array}{l}\text{ 定义 : }\left(\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\right)=\left(\overrightarrow{a}\times \overrightarrow{b}\right)\cdot \overrightarrow{c}\left(\text{ 其中 }\overrightarrow{a}=\left\{x_1,y_1,z_1\right\},\overrightarrow{b}=\left\{x_2,y_2,z_2\right\},\overrightarrow{c}=\left\{x_3,y_3,z_3\right\}\right)\\ \text{ 公式 : }\left(\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\right)=\left|\begin{array}{lll}{x}_1& y_1& z_1\\ x_2& y_2& z_2\\ x_3& y_3& z_3\end{array}\right|\end{array}$$

2. 混合积的几何意义:

$$\{\begin{array}{l}\text{以}\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\text{为棱的平行六面体体积}=\left|\left(\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\right)\right|\\ \overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\text{三向量共面,则混合积}\left(\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\right)=0\end{array}$$

第 3 组: 向量的运算规律

1. 数量积:

$$\{\begin{array}{l}\text{ 交换 : }\overrightarrow{a}\cdot \overrightarrow{b}=\overrightarrow{b}\cdot \overrightarrow{a}\\ \text{ 分配 : }\overrightarrow{a}\cdot \left(\overrightarrow{b}+\overrightarrow{c}\right)=\overrightarrow{a}\cdot \overrightarrow{b}+\overrightarrow{a}\cdot \overrightarrow{c}\end{array}$$

2. 向量积:

$$\{\begin{array}{l}\text{ 交换: }\overrightarrow{a}\times \overrightarrow{b}=-\left(\overrightarrow{b}\times \overrightarrow{a}\right)\\ \text{ 分配: }\overrightarrow{a}\times \left(\overrightarrow{b}+\overrightarrow{c}\right)=\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{a}\times \overrightarrow{c}\end{array}$$

3. 混合积:

$$\{\begin{array}{l}\text{ 次序不变: }\left(\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\right)=\left(\overrightarrow{b},\overrightarrow{c},\overrightarrow{a}\right)=\left(\overrightarrow{c},\overrightarrow{a},\overrightarrow{b}\right)\\ \text{ 次序改变: }\left(\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\right)=-\left(\overrightarrow{a},\overrightarrow{c},\overrightarrow{b}\right)\end{array}$$

第 4 组: 空间单位向量

单位向量:

$$\{\begin{array}{l}\overrightarrow{a}=\{x,y,z\}\text{ 对应的单位向量为 }\{\cos \alpha ,\cos \beta ,\cos \gamma \}\\ \cos \alpha =\frac{x}{\sqrt{x^2+y^2+z^2}},\cos \beta =\frac{y}{\sqrt{x^2+y^2+z^2}},\cos \gamma =\frac{z}{\sqrt{x^2+y^2+z^2}}.\end{array}$$

性质: ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$