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yabo4:[教案精品]新课标高中数学人教A版必修四全册教案2.4.2平面向量数量积的坐标表示、模、夹角

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2.4.2 平面向量数量积的坐标表示、模、夹角
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
2

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| 或 | a |?

a ?a

4?cos? =

a?b ; | a || b |

5?|a?b| ≤ |a||b|

3.练习: (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

? ,那么向量 m=a-4b 的模为( 3
D.12

A.2 二、讲解新课:

B.2 3

C.6

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

x2 ? y2 .

1

(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 ) 、 ( x 2 , y 2 ) , 那么 | a |?

( x1 ? x 2 ) 2 ? ( 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 4. 两向量夹角的余弦( 0 ? ? ? ? ) cos? =

a ?b ? | 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 1 )
o

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 2、 已知 A(3, B(-1, 2), -1), 若点 P(x, 四、小结: 1、 a ? b ? x1 x2 ? y1 y 2 2、平面内两点间的距离公式 | a |? 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。

1 )在线段 AB 的中垂线上, x= 则 2

.

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

2

思考: 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:x + y ?5x ? 2y = 0
2 2

又∵| OB | = | AB |

∴x + y = (x?5) + (y?2) 即:10x + 4y = 29
2 2 2 2

? 7 3 ? ? x 2 ? y 2 ? 5 x ? 2 y ? 0 ? x1 ? 2 ? x2 ? 2 ? 由? ?? 或 3 ? 7 ?10 x ? 4 y ? 29 ? y1 ? ? ? y2 ? ? 2 ? 2 ?
∴B 点坐标 ( ,? ) 或 ( , ) ; AB = (?

7 2

3 2

3 7 2 2

3 7 7 3 ,? ) 或 (? , ) 2 2 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 ∴k =

11 3
∴k =

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

3 ? 13 2

3


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