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huangguancaipiaodaohang:2019年高中数学 2.4.2平面向量数量积的坐标表示、模、夹角教案 新人教A版必修4

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2019 年高中数学 2.4.2 平面向量数量积的坐标表示、模、夹角教案 新人教 A 版必修 4
教学目的: 1.掌握平面向量数量积运算规律; 2.能利用数量积的 5 个重要性质及数量积运算规律解决有关问题; 3.掌握两个向量共线、垂直的几何判断,会证明两向量垂直,以及能解决一些简单问题. 教学重点:平面向量数量积及运算规律. 教学难点:平面向量数量积的应用 教学过程: 一、复习引入: 1.平面向量数量积(内积)的定义: 2.两个向量的数量积的性质: 设 a、b 为两个非零向量,e 是与 b 同向的单位向量. 1? e?a = a?e =|a|cos?; 2? a?b ? a?b = 0 3? 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 4?cos? = | a || b | ;
3.练习:

5?|a?b| ≤ |a| |b|

(1)已知|a|=1,|b|= 2 ,且(a-b)与 a 垂直,则 a 与 b 的夹角是( A.60° B.30° C.135° D.45°



? (2)已知|a|= 2,|b|=1,a 与 b 之间的夹角为 3 ,那么向量 m=a-4b 的模为(
A.2 二、讲解新课: B.2 3 C.6 D.12



探究:已知两个非零向量 a ? ( x1 , y1 ) , b ? ( x2 , y 2 ) ,怎样用 a 和 b 的坐标表示 a ? b ?. 1、平面两向量数量积的坐标表示 两个向量的数量积等于它们对应坐标的乘积的和.即 a ? b ? x1 x2 ? y1 y 2 2. 平面内两点间的距离公式

| a |? (1)设 a ? ( x, y) ,则 | a | ? x ? y 或
2 2 2

x2 ? y2

.

(2)如果表示向量 a 的有向线段的起点和终点的坐标分别为 ( x1 , y1 ) 、 ( x2 , y 2 ) , 那么

| a |? ( x1 ? x 2 ) 2 ? ( y1 ? y 2 ) 2

(平面内两点间的距离公式)

向量垂直的判定

设 a ? ( x1 , y1 ) , b ? ( x2 , y 2 ) ,则 a ? b ? x1 x2 ? y1 y 2 ? 0 两向量夹角的余弦( 0 ? ? ? ? )

a ?b ? co s? = | a | ? | b |

x1 x 2 ? y1 y 2 x1 ? y1
2 2

x2 ? y2

2

2

二、讲解范例: 例 1 已知 A(1, 2),B(2, 3),C(?2, 5),试判断△ABC 的形状,并给出证明. 例 2 设 a = (5, ?7),b = (?6, ?4),求 a·b 及 a、b 间的夹角 θ (精确到 1o) 分析:为求 a 与 b 夹角,需先求 a·b 及|a|·|b|,再结合夹角 θ 的范围确定其值. 例 3 已知 a=(1, 3 ) ,b=( 3 +1, 3 -1) ,则 a 与 b 的夹角是多少? 分析:为求 a 与 b 夹 角,需先求 a·b 及|a|·|b|,再结合夹角 θ 的范围确定其值. 解:由 a=(1, 3 ) ,b=( 3 +1, 3 -1) 有 a·b= 3 +1+ 3 ( 3 -1)=4,|a|=2,|b|=2 2 .

记 a 与 b 的夹角为 θ ,则cosθ =

a?b 2 ? a?b 2

? 又∵0≤θ ≤π ,∴θ = 4

评述:已知三角形函数值求角时,应 注重角的范围的确定. 三、课堂练习:1、P107 面 1、2、3 题

1 2、 已知 A(3, 2), B(-1, -1), 若点 P(x, - 2 )在线段 AB 的中垂线上, 则 x=
四、小结: 1、 a ? b ? x1 x2 ? y1 y 2 2、平面内两点间的距离公式 3、向量垂直的判定: 设 a ? ( x1 , y1 ) , b ? ( x2 , y 2 ) ,则 a ? b ? x1 x2 ? y1 y 2 ? 0 五、课后作业: 《习案》作业二十四。 思考:

.

| a |? ( x1 ? x 2 ) 2 ? ( y1 ? y 2 ) 2

1、如图,以原点和 A(5, 2)为顶点作等腰直角△OAB,使?B = 90?,求点 B 和向量 AB 的 坐标. 解:设 B 点坐标(x, y),则 OB = (x, y), AB = (x?5, y?2) ∵ OB ? AB ∴ x(x?5) + y(y?2) = 0 即:x2 + y2 ?5x ? 2y = 0 ∴x2 + y2 = (x?5)2 + (y?2)2 即:10x + 4y = 29

又∵| OB | = | AB |

? 7 3 ? x1 ? x2 ? ?x 2 ? y 2 ? 5x ? 2 y ? 0 ? ? ? 2 或 2 ?? ? ? 3 7 10 x ? 4 y ? 29 ? ? y1 ? ? ? y2 ? ? 2 ? 2 ? 由
7 3 3 7 3 7 7 3 ( ,? ) ( , ) ( ? ,? ) ( ? , ) ∴B 点坐标 2 2 或 2 2 ; AB = 2 2 或 2 2
2 在△ABC 中, AB =( 2, 3), AC =(1, k),且△ABC 的一个内角为直角,求 k 值.

解:当 A = 90?时, AB ? AC = 0,∴2×1 +3×k = 0

?
∴k =

3 2

当 B = 90?时, AB ? BC = 0, BC = AC ? AB = (1?2, k?3) = (?1, k?3)

∴2×(?1) +3×(k?3) = 0

11 ∴k = 3

当 C = 90?时, AC ? BC = 0 ,∴?1 + k(k?3) = 0

3 ? 13 2 ∴k =


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