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https://www.lanzous.com:高一数学精品课件2.4.1 平面向量数量积的物理背景及其含义 课件_图文

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第二章 平面向量 2.4 平面向量的数量积 2.4.1 平面向量数量积的物理背景及 其含义 学习导航 学习目标 平面向量的数量积 理解 实例 ― ― → 与向量投影的关系 ― ― → 了解 平面向量数量积的 掌握 平面向量数量积的 ― → 重要性质及运算律 含义及物理意义 ― 重点难点 重点:平面向量数量积概念、运算律及其相 关性质和运用. 难点:数量积的几何意义. 新知初探思维启动 1.平面向量数量积的定义 已知两非零向量 a 与 b ,它们的夹角为 θ ,则把数量 |a||b|· cos θ 叫做a与b的_________ 内积 , 数量积 (或______) ___________ a· b=|a||b|cos θ. 记作a· b,即______________ 0 规定零向量与任一向量的数量积均为______. 想一想 1.向量的数量积与向量的数乘相同吗? 提示:不相同.向量的数量积a· b是一个实数;数乘向量λa是一 个向量. 做一做 1.若|m|=4,|n|=6,m与n的夹角为135°,则m· n=________. 解析: m· n=|m||n|cos 135° =4×6×?- ? 2? =-12 2. 2? 答案:-12 2 2.向量的数量积的几何意义 (1)投影:|a|cos θ(|b|cos θ)叫做向量a在b方向上(b在a方向 投影. 上)的________ (2) 几何意义:数量积 a· b 等于 a 的长度 |a| 与 b 在 a 的方向上 |b|cos θ 的乘积. 的投影___________ 想一想 2.投影是向量吗? 提示:投影是数量而不是向量,它可正可负可为零,它 的符号由θ的取值决定. 3.向量的数量积的性质 设 a 与 b 都是非零向量, θ 为 a 与 b 的夹角. a· b= 0 (1)a⊥b?_____________. (2)当 a 与 b 同向时, a· b=_________, |a||b| -|a||b| 当 a 与 b 反向时, a· b=__________. (3)a· a=______或 |a|= a· a= a2. (4)cos a· b |a||b| θ= __________. |a|2 ≤ a||b|. (5)|a· b|_______| 做一做 2.已知|a|=5,|b|=4,a· b=10 3,则 a 与 b 的夹角 θ= ________. 答案:30° 4.向量数量积的运算律 b· a (交换律). (1)a· b=_______ λ(a· b)=a· (λb) 结合律). (2)(λa)· b=_______________( a· c+b· c (3)(a+b)· c=______________ (分配律). 想一想 3.对于向量a· b· c,等式(a· b)· c=a· (b· c)一定成立吗? 提示:不一定成立,∵若(a· b)· c≠0,其方向与c相同或 相反,而a· (b· c)≠0时其方向与a相同或相反,而a与c方 向不一定相同,故该等式不一定成立. 典题例证技法归纳 题型探究 题型一 向量数量积的运算 例1 (1)已知 |a|=4,|b|=5,且向量 a 与 b 的夹角为 60° , 求(2a+ 3b)· (3a-2b); → → (2)在 Rt△ABC 中,∠ C=90° , AB= 5,AC=4,求AB· BC. 【解】 (1)(2a+ 3b)· (3a-2b) = 6a2- 4a· b+9a· b- 6b2 = 6× 42+5×4× 5× cos 60° -6× 52=- 4. (2)在 Rt△ ABC 中,∠ C=90° , AB= 5, 3 AC= 4,故 BC=3,且 cos∠ ABC= , 5 → → AB与BC的夹角 θ=180° -∠ ABC, 3 → → → → ∴AB· BC=- |AB||BC|cos∠ ABC=- 5×3× =- 9. 5 【名师点评】 求两向量数量积的步骤是: (1)求a与b的夹角; (2)分别求|a|,|b|; (3)求数量积,即a· b=|a||b|cos θ.应注意书写时a与b之间用 “· ”连接,而不能用“×”连接,也不能省去. 跟踪训练 1.已知|a|=3,|b|=6,当①a∥b,②a⊥b,③a 与b的夹角是60°时,分别求a· b. 解:①当a∥b时,若a与b同向,则它们的夹角θ=0°, ∴ a· b=|a|· |b|cos 0°=3×6×1=18; 若a与b反向,则它们的夹角θ=180°, ∴a· b=|a||b|cos 180°=3×6×(-1)=-18; ②当 a⊥ b 时,它们的夹角 θ= 90° , ∴ a· b= 0; ③当 a 与 b 的夹角是 60° 时,有 a· b= |a||b|cos 60° 1 = 3× 6× = 9. 2 题型二 例2 向量的模长问题 π (1)已知|a|=|b|=5,向量 a 与 b 的夹角 θ 为 , 3 则|a+b|=________. (2)(2012· 高考课标全国卷)已知向量 a, b 夹角为 45° , 且|a| =1,|2a-b|= 10,则|b|=________. 【解析】 |a+ b|= = 1 25 (1)a· b= |a||b|cos θ= 5× 5× = . 2 2 ?a+b? 2= |a|2+ 2a· b+ |b|2 25 25+ 2× + 25= 5 3. 2 (2)把 |2a- b|= 10平方得 4|a|2-4|a|· |b|cos 45° + |b|2= 10.∵ |a|=1,∴ |b|2- 2 2|b|- 6=0. ∴ |b|= 3 2或 |b|=- 2(舍去 ). 【答案】 (1)5 3 (2)3 2 【名师点评】 (1)此类求解模问题一般转化为

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