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www.cpcp97.com:高中数学第二章平面向量2.4平面向量的数量积的物理背景及其含义教案新人教A版必修4

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课题: 平面向量的数量积的物理背景及其含义

[课时安排]

1 课时

[教学目标]

1.知识与技能: 掌握平面向量的数量积及其几何意义;掌

握平面向量数量积的重要性质及运算律;了解用平面向量的

数量积可以处理有关长度、角度和垂直的问题

2.过程与方法: 启发式教学,引导学生思路

3.情感、态度与价值观: 经历由实际问题建立数学模型的

过程,体会其基本方法.

[教学重点]

平面向量的数量积定义及应用

[教学难点]

平面向量数量积的定义及运算律的理解

[教学器材]

[教法学法]

[教学过程] 一、复习准备: 1. 如何由坐标得到两个向量共线? 2. 物理中力做的功是怎样定义的? 二、讲授新课: 1.教学向量的数量积的概念.

备注

①.两个非零向量夹角的概念:已知非零向量a与b,作 OA =a, OB =b,

则∠AOB=θ (0≤θ ≤π )叫a与b的夹角.注意:当 θ =0时a与b同

向;当 θ =π 时,a与b反向;当 θ = ? 时,a与b垂直,记a⊥b; 2
②.平面向量数量积(内积)的定义: 已知两个非零向量a与b,它们的夹角

是θ ,则数量|a||b

叫a与b的数量积,记作 a b,即有 a b =

|a||b



(分析:符号由

的符号所决定;两个向量的数量积称为内积,写成 a b;)

③.“投影”的概念:作图定义:|b

叫做向量 b 在 a 方向上的投影. 投

影也是一个数量,不是向量;当 为锐角时投影为正值;当 为钝角时投影为

负值;当 为直角时投影为 0;当 = 0 时投影为 |b|;当 = 180 时投

影为 b|

1

小中高学习资料 推荐下载

④.向量的数量积的几何意义:数量积 a b 等于 a 的长度与 b 在 a 方向上投

影|b| 的乘积.

⑤.性质:e a = a e =|a

,a b a b = 0,当 a 与 b 同向时,

a b = |a||b|;当 a 与 b 反向时,a b

a||b|. 特别的 a a = |a|2 或

| a |? a ? a

= a?b )
| a || b |

⑥探究:运算律 a b=b.a (λ a).b=λ (a.b)

2.教学例题

①.讲解范例:例 1 已知|a|=5, |b|=4, a 与 b 的夹角 θ =120o,求 a·b.

例 2 已知|a|=6, |b|=4, a 与 b 的夹角为 60o 求(a+2b)·(a-3b).

例 3 已知|a|=3, |b|=4, 且 a 与 b 不共线,k 为何值时,向量 a+kb 与 a-kb

互相垂直.

(教师演示 ? 学生模仿 ? 学生演示)

②.练习: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.

3. 小结:1.平面向量数量积(内积)的定义;2.向量的数量积的几何意义

[教学反思]

养成良好的学习惯,有利于激发生积极性和主动;形策略提高效率培自能力创新精神造使终身受益Mr.Johnsadevbupifltc,在教师讲课之前己先独立地阅读内容。初步理解是上做接知识准备过程些由没预对老一堂要无所坐等wygF加油就会功命不息

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