高中数学北师大版必修第二册“数量积与垂直关系”核心知识清单

高中数学北师大版必修第二册“数量积与垂直关系”核心知识清单一、 【基础脉络】向量的数量积：从定义到本质的理解（一） 向量数量积的源头定义——不仅仅是公式，更是投影的积1. 【基础】概念的精确定义：对于两个非零向量a⃗\vec{a}a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">与b⃗\vec{b}b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">，它们的夹角为θ\thetaθ（θ∈[0,π]\theta\in[0,\pi]θ∈[0,π]），则数量∣a⃗∣∣b⃗∣cos⁡θ|\vec{a}||\vec{b}|\cos\theta∣a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">∣∣b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">∣cosθ叫做a⃗\vec{a}a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">与b⃗\vec{b}b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">的数量积（或内积、点积），记作a⃗⋅b⃗\vec{a}\cdot\vec{b}a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">⋅b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">。即：a⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\thetaa<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">⋅b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">=∣a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">∣∣b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">∣cosθ。特别规定：零向量与任一向量的数量积为0。2. 【重要】几何意义的深度剖析：数量积的几何意义是“投影的乘积”。具体而言，a⃗⋅b⃗\vec{a}\cdot\vec{b}a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">⋅b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">等于a⃗\vec{a}a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">的长度∣a⃗∣|\vec{a}|∣a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">∣与b⃗\vec{b}b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">在a⃗\vec{a}a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">方向上的投影∣b⃗∣cos⁡θ|\vec{b}|\cos\theta∣b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">∣cosθ的乘积；也等于b⃗\vec{b}b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">的长度∣b⃗∣|\vec{b}|∣b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">∣与a⃗\vec{a}a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">在b⃗\vec{b}b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">方向上的投影∣a⃗∣cos⁡θ|\vec{a}|\cos\theta∣a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">∣cosθ的乘积。这揭示了数量积将两个向量的长度和它们的相对方向（夹角）联系起来。3. 【易错澄清】夹角θ\thetaθ必须是两个向量“同起点”时所夹的角。若两个向量起点不一致，应通过平移使其起点重合后再确定夹角。例如，在三角形ABCABCABC中，求AB⃗⋅BC⃗\vec{AB}\cdot\vec{BC}AB<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">⋅BC<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">时，应将AB⃗\vec{AB}AB<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">与BC⃗\vec{BC}BC<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">平移至起点重合，发现其夹角并非内角BBB，而是π−B\piBπ−B。（二） 投影与投影向量——从数量到向量的升华1. 【基础】投影（数量）的概念：向量b⃗\vec{b}b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">在a⃗\vec{a}a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">方向上的投影是一个实数，计算公式为∣b⃗∣cos⁡θ|\vec{b}|\cos\theta∣b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">∣cosθ。它可正、可负、可为零，其正负由夹角θ\thetaθ决定：锐角时为正，钝角时为负，直角时为零。2. 【拓展】投影向量：向量b⃗\vec{b}b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">在a⃗\vec{a}a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">方向上的投影向量是一个向量，其方向与a⃗\vec{a}a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">同向或反向。若设与a⃗\vec{a}a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">方向相同的单位向量为e⃗\vec{e}e<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">，则b⃗\vec{b}b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">在a⃗\vec{a}a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">上的投影向量为(∣b⃗∣cos⁡θ)e⃗(|\vec{b}|\cos\theta)\vec{e}(∣b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">∣cosθ)e<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">。它的模长即为投影的绝对值。（三） 【高频考点】数量积的物理背景——功的运算1. 物理模型：在物理学中，一个恒力F⃗\vec{F}F<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">作用在物体上，使物体产生位移s⃗\vec{s}s<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">，则力F⃗\vec{F}F<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">所做的功WWW即为F⃗\vec{F}F<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">与s⃗\vec{s}s<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">的数量积：W=F⃗⋅s⃗=∣F⃗∣∣s⃗∣cos⁡θW=\vec{F}\cdot\vec{s}=|\vec{F}||\vec{s}|\cos\thetaW=F<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">⋅s<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">=∣F<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">∣∣s<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">∣cosθ（θ\thetaθ为力与位移的夹角）。这为我们理解数量积提供了直观的物理背景：它衡量了一个向量在另一个向量方向上的作用效果。二、 【核心运算律】数量积的代数性质与易错警示（一） 【基础】数量积的运算律1. 交换律：a⃗⋅b⃗=b⃗⋅a⃗\vec{a}\cdot\vec{b}=\vec{b}\cdot\vec{a}a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">⋅b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">=b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">⋅a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">。2. 数乘结合律：(λa⃗)⋅b⃗=λ(a⃗⋅b⃗)=a⃗⋅(λb⃗)(\lambda\vec{a})\cdot\vec{b}=\lambda(\vec{a}\cdot\vec{b})=\vec{a}\cdot(\lambda\vec{b})(λa<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">)⋅b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">=λ(a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">⋅b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">)=a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">⋅(λb<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">)，其中λ∈R\lambda\in\mathbb{R}λ∈R。3. 分配律：(a⃗+b⃗)⋅c⃗=a⃗⋅c⃗+b⃗⋅c⃗(\vec{a}+\vec{b})\cdot\vec{c}=\vec{a}\cdot\vec{c}+\vec{b}\cdot\vec{c}(a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">+b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">)⋅c<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">=a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">⋅c<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">+b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">⋅c<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">。这是向量运算转化为代数运算的桥梁，必须熟练掌握。（二） 【难点·必纠】数量积运算的“三大禁区”1. 【易错】不能“约分”：由a⃗⋅b⃗=a⃗⋅c⃗\vec{a}\cdot\vec{b}=\vec{a}\cdot\vec{c}a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">⋅b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">=a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">⋅c<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">且a⃗≠0⃗\vec{a}\neq\vec{0}a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">=0<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">，不能推出b⃗=c⃗\vec{b}=\vec{c}b<pathd="M37720c05.3331..514Sc4.6670