8. 向量的投影与内积
复习前面的知识：，若BCE三点共线，则
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\vec{AE}=(1-s)\vec{AB}+s\vec{AC},(B,C,E)=\mu\Rightarrow s=\frac{\mu}{1+\mu},1-s=\frac{1}{1+\mu}
AE=(1−s)AB+sAC,(B,C,E)=μ⇒s=1+μμ,1−s=1+μ1
Ceva定理的前提条件是E F D是三角形内点，假设E不是三角形内点
8.1 向量的投影
- 给定一个非零向量 e \pmb{e} e， α \pmb{\alpha} α可分解为 α 1 + α 2 \pmb{\alpha}_{1}+\pmb{\alpha}_{2} α1+α2使得 α 1 / / e , α 2 ⊥ e \pmb{\alpha}_{1}//\pmb{e},\pmb{\alpha}_{2}\bot\pmb{e} α1//e,α2⊥e，且分解系数唯一。将 α 1 \pmb{\alpha}_{1} α1称为 α \pmb{\alpha} α关于向量 e \pmb{e} e的内投影，记为 P e α P_{\pmb{e}}\pmb{\alpha} Peα， α 2 \pmb{\alpha}_{2} α2称为 α \pmb{\alpha} α关于向量 e \pmb{e} e的外投影，记为 P ˉ e α \bar{P}_{\pmb{e}}\pmb{\alpha} Pˉeα；
- P e α , P ˉ e α P_{\pmb{e}}\pmb{\alpha},\bar{P}_{\pmb{e}}\pmb{\alpha} Peα,Pˉeα与 e \pmb{e} e的大小无关，只与 e \pmb{e} e的方向有关。
- 对两个非零向量 α \pmb{\alpha} α与 β \pmb{\beta} β的几何夹角记作 < α , β > <\pmb{\alpha},\pmb{\beta}> <α,β>，其中 ( 0 ∘ ≤ ( < α , β > ) < 18 0 ∘ (0^{\circ}\le(<\pmb{\alpha},\pmb{\beta}>)<180^{\circ} (0∘≤(<α,β>)<180∘
- 记 e 0 = e ∣ e ∣ \pmb{e}_{0}=\frac{\pmb{e}}{|\pmb{e}|} e0=∣e∣e，当 α ≠ 0 ⃗ , P e α = ∣ α ∣ cos < α , β > e 0 \pmb{\alpha}\ne\vec{0},P_{\pmb{e}}\pmb{\alpha}=|\pmb{\alpha}|\cos<\pmb{\alpha},\pmb{\beta}>\pmb{e}_{0} α=0,Peα=∣α∣cos<α,β>e0
- 【内投影的线性性】
（1） P e ( α + β ) = P e α + P e β P_{\pmb{e}}(\pmb{\alpha}+\pmb{\beta})=P_{\pmb{e}}\pmb{\alpha}+P_{\pmb{e}}\pmb{\beta} Pe(α+β)=Peα+Peβ
P ˉ e ( α + β ) = P ˉ e α + P e β \bar{P}_{\pmb{e}}(\pmb{\alpha}+\pmb{\beta})=\bar{P}_{\pmb{e}}\pmb{\alpha}+P_{\pmb{e}}\pmb{\beta} Pˉe(α+β)=Pˉeα+Peβ
（2） P e ( λ α ) = λ P e α , λ ∈ R P_{\pmb{e}}(\lambda\pmb{\alpha})=\lambda P_{\pmb{e}}\pmb{\alpha},\lambda\in\mathbb{R} Pe(λα)=λPeα,λ∈R
P ˉ e ( λ α ) = λ P ˉ e α , λ ∈ R \bar{P}_{\pmb{e}}(\lambda\pmb{\alpha})=\lambda \bar{P}_{\pmb{e}}\pmb{\alpha},\lambda\in\mathbb{R} Pˉe(λα)=λPˉeα,λ∈R
【证】由内投影和外投影的定义可知， α = P e α + P ˉ e α \pmb{\alpha}=P_{\pmb{e}}\pmb{\alpha}+\bar{P}_{\pmb{e}}\pmb{\alpha} α=Peα+Pˉeα
β = P e β + P ˉ e β \pmb{\beta}=P_{\pmb{e}}\pmb{\beta}+\bar{P}_{\pmb{e}}\pmb{\beta} β=Peβ+Pˉeβ
则 α + β = P e α + P ˉ e α + P e β + P ˉ e β = ( P e α + P e β ) + ( P ˉ e α + P ˉ e β ) \pmb{\alpha}+\pmb{\beta}=P_{\pmb{e}}\pmb{\alpha}+\bar{P}_{\pmb{e}}\pmb{\alpha}+P_{\pmb{e}}\pmb{\beta}+\bar{P}_{\pmb{e}}\pmb{\beta}=(P_{\pmb{e}}\pmb{\alpha}+P_{\pmb{e}}\pmb{\beta})+(\bar{P}_{\pmb{e}}\pmb{\alpha}+\bar{P}_{\pmb{e}}\pmb{\beta}) α+β=Peα+Pˉeα+Peβ+Pˉeβ=(Peα+Peβ)+(Pˉeα+Pˉeβ)（分解成了一个垂直于 e \pmb{e} e的向量和一个平行于 e \pmb{e} e的向量相加）
由投影分解的唯一性， P e ( α + β ) = P e α + P e β , P ˉ e ( α + β ) = P ˉ e α + P e β P_{\pmb{e}}(\pmb{\alpha}+\pmb{\beta})=P_{\pmb{e}}\pmb{\alpha}+P_{\pmb{e}}\pmb{\beta},\bar{P}_{\pmb{e}}(\pmb{\alpha}+\pmb{\beta})=\bar{P}_{\pmb{e}}\pmb{\alpha}+P_{\pmb{e}}\pmb{\beta} Pe(α+β)=Peα+Peβ,Pˉe(α+β)=Pˉeα+Peβ（对应项相等）
8.2 向量的内积
定义：有向量 α , β \pmb{\alpha},\pmb{\beta} α,β， α ⋅ β = ∣ α ∣ ∣ β ∣ ⋅ cos < α , β > \pmb{\alpha}\cdot\pmb{\beta}=|\pmb{\alpha}||\pmb{\beta}|\cdot\cos<\pmb{\alpha},\pmb{\beta}> α⋅β=∣α∣∣β∣⋅cos<α,β>（结果是一个数）
- α ⋅ β = 0 ⇔ α ⊥ β \pmb{\alpha}\cdot\pmb{\beta}=0\Leftrightarrow\pmb{\alpha}\bot\pmb{\beta} α⋅β=0⇔α⊥β
- α ⋅ α = ∣ α ∣ 2 ⇒ α ⋅ α \pmb{\alpha}\cdot\pmb{\alpha}=|\pmb{\alpha}|^{2}\Rightarrow\sqrt{\pmb{\alpha}\cdot\pmb{\alpha}} α⋅α=∣α∣2⇒α⋅α
- < α , β > = arccos α ⋅ β ∣ α ∣ ∣ β ∣ <\pmb{\alpha},\pmb{\beta}>=\arccos\frac{\pmb{\alpha}\cdot\pmb{\beta}}{|\pmb{\alpha}||\pmb{\beta}|} <α,β>=arccos∣α∣∣β∣α⋅β
- 【交换律】 α ⋅ β = β ⋅ α \pmb{\alpha}\cdot\pmb{\beta}=\pmb{\beta}\cdot\pmb{\alpha} α⋅β=β⋅α
- P β α = ∣ α ∣ cos < α , β > β 0 = ∣ α ∣ cos < α , β > β ∣ β ∣ ⇒ α ⋅ β = P β α β 0 ∣ β ∣ P_{\pmb{\beta}}\pmb{\alpha}=|\alpha|\cos<\pmb{\alpha},\pmb{\beta}>\pmb{\beta}_{0}=|\alpha|\cos<\pmb{\alpha},\pmb{\beta}>\frac{\pmb{\beta}}{|\pmb{\beta}|}\Rightarrow\pmb{\alpha}\cdot\pmb{\beta}=\frac{P_{\pmb{\beta}}\pmb{\alpha}}{\pmb{\beta}_{0}}|\pmb{\beta}| Pβα=∣α∣cos<α,β>β0=∣α∣cos<α,β>∣β∣β⇒α⋅β=β0Pβα∣β∣
- 【定理1.3】内积具有双线性性
（1） α ⋅ ( β + γ ) = α ⋅ β + α ⋅ γ \pmb{\alpha}\cdot(\pmb{\beta}+\pmb{\gamma})=\pmb{\alpha}\cdot\pmb{\beta}+\pmb{\alpha}\cdot\pmb{\gamma} α⋅(β+γ)=α⋅β+α⋅γ
( α + γ ) ⋅ β = α ⋅ β + γ ⋅ α (\pmb{\alpha}+\pmb{\gamma})\cdot\pmb{\beta}=\pmb{\alpha}\cdot\pmb{\beta}+\pmb{\gamma}\cdot\pmb{\alpha} (α+γ)⋅β=α⋅β+γ⋅α
（2） α ⋅ ( λ β ) = λ α β = ( λ α ) β \pmb{\alpha}\cdot(\lambda \pmb{\beta})=\lambda \pmb{\alpha}\pmb{\beta}=(\lambda\pmb{\alpha})\pmb{\beta} α⋅(λβ)=λαβ=(λα)β
