线性代数2

向量可以平移，向量模长与方向确定也不唯一
向量使矢量，既有大小又有方向

数量积：

$$\begin{gathered} \vec{a}\cdot \vec{b}=|\vec{a}|\ast |\vec{b}|\ast cos\theta \\ \vec{a}\cdot \vec{b}=x_1{y}_1+x_2{y}_2+\cdots +x_n{y}_n \end{gathered}$$

$$\begin{gathered} |\vec{c}{|}^2=(|\vec{a}|sin\theta {)}^2+(|\vec{b}|-|\vec{a}|cos\theta {)}^2 \\ |\vec{c}{|}^2=|\vec{a}{|}^{2}si{n}^2\theta +|\vec{b}{|}^2-2|\vec{b}||\vec{a}|cos\theta +|\vec{a}{|}^{2}co{s}^2\theta \\ |\vec{c}{|}^2=|\vec{a}{|}^2(si{n}^2\theta +co{s}^2\theta )+|\vec{b}{|}^2-2|\vec{b}||\vec{a}|cos\theta \\ si{n}^2\theta +co{s}^2\theta =1 \\ |\vec{c}{|}^2=|\vec{a}{|}^2+|\vec{b}{|}^2-2|\vec{b}||\vec{a}|cos\theta \end{gathered}$$
坐标表示：$$\begin{gathered} \vec{c}=\vec{a}-\vec{b} \\ \vec{a}-\vec{b}=(x_1-x_2,y_1-y_2) \\ |\vec{c}{|}^2=(x_1-x_2{)}^2+(y_1-y_2{)}^2 \\ |\vec{a}{|}^2=x_1^2+y_1^2 \\ |\vec{b}{|}^2=x_2^2+y_2^2 \end{gathered}$$$$\begin{gathered} |\vec{c}{|}^2=|\vec{a}{|}^2+|\vec{b}{|}^2-2|\vec{b}||\vec{a}|cos\theta \\ (x_1-x_2{)}^2+(y_1-y_2{)}^2=x_1^2+y_1^2+x_2^2+y_2^2-2|\vec{b}||\vec{a}|cos\theta \\ {x}_1^2-2x_1{x}_2+x_2^2+y_1^2-2y_1{y}_2+y_2^2=x_1^2+y_1^2+x_2^2+y_2^2-2|\vec{b}||\vec{a}|cos\theta \\ {\bcancel{x}}_1^2-2x_1{x}_2+{\bcancel{x}}_2^2+{\bcancel{y}}_1^2-2y_1{y}_2+{\bcancel{y}}_2^2={\bcancel{x}}_1^2+{\bcancel{y}}_1^2+{\bcancel{x}}_2^2+{\bcancel{y}}_2^2-2|\vec{b}||\vec{a}|cos\theta \\ -2x_1{x}_2-2y_1{y}_2+=-2|\vec{b}||\vec{a}|cos\theta \\ 2x_1{x}_2+2y_1{y}_2=2|\vec{b}||\vec{a}|cos\theta \\ {x}_1{x}_2+y_1{y}_2=|\vec{b}||\vec{a}|cos\theta \\ |\vec{b}||\vec{a}|cos\theta =x_1{x}_2+y_1{y}_2 \end{gathered}$$

$$\begin{gathered} \vec{a}\cdot \vec{b}=|\vec{a}|\ast |\vec{b}|\ast cos\theta \\ |\vec{b}||\vec{a}|cos\theta =x_1{x}_2+y_1{y}_2 \\ \vec{a}\cdot \vec{b}=x_1{x}_2+y_1{y}_2 \end{gathered}$$

向量积

$$|\vec{a}\times \vec{b}|=|\vec{a}|\ast |\vec{b}|\ast sin\theta$$

矩阵

矩阵，即描述线性代数中线性关系的参数，即矩阵是一个线性变换，可以将一些变量转换为另一些向量
矩阵是向量的集合