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www.45966.com:高中数学 第二章 平面向量 2.4 向量的数量积(2)教案 苏教版必修4_图文

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高中数学 必修4

问题情境

【复习提问】
1. (1)两个非零向量夹角的概念;
(2)平面向量数量积的定义; (3)“投影”的概念;
(4)向量数量积的几何意义; (5)两个向量的数量积的性质.
2.判断下列各题正确与否:

①若 a ? 0,则对任一向量 b, a ?b ? 0 ②若a ? 0,则对任一向量 b , a ?b ? 0

() ()

③若 a ? 0, a ?b ? 0 ,则 b ? 0

()

④若 a ?b ? 0 ,则 a, b 至少有一个为零向量; ( ) ⑤若a ?b ? a ? c,则 b ? c当且仅当 a ? 0 时成立; ( )

⑥对任意向量

a

,有

2
a

?|

a

|2

()

学生活动
【提出问题】已知实数 a 、b、c ( b ? 0 ),则 a ? b ? b ? c ? a ? c
a ? b =b·c ? a = c 是否成立?
【提出问题】实数的运算律有ab=ba;a(b+c)=ab+ac;(ab)c=a(bc). 在向量的数量积中是否成立?(举例说明)

建构数学

数量积的运算律

(1)交换律:a ?b ? b ? a

(2)数乘结合律:(?a) ?b ? ?(a ?b) ? a ? (?b)

(3)分配律:(a ? b) ? c ? a ? c ? b ? c

【说明】

? (1)一般地,

? a

?

b???

? c

?

? a

?

?b?

?

? c

?

(2)

? a

?

? c

?

? b

?

? c,

? c

?

? 0

? a

?

? b

(3)有如下常用性质

? ? ?
a

2

?

? a

2

? a

? ?b

2

?

? a

2

?

? 2a

? ?b

? ?b2

? ? ?a?

?

? c

??

?? b ?d

?? ? ? ?? ? ? ? a?b ?a?d ?b ?c ?c ?d

【例题讲解】
例1 已知 a, b 都是非零向量,且 a ? 3b与 7a ? 5b 垂直,
a ? 4b 与 7a ? 2b垂直,求 a与 b 的夹角.

例2 求证:平行四边形两条对角线平方和等于四条边的平方和.

变式1:用向量方法证明:菱形对角线互相垂直.
A

E
变式2. 如图,AD, BE,CF 是 ?ABC 的三条高, F H

求证:AD, BE,CF相交于一点.

B

D

C

变式3:用向量证明三角形的三条角平分线相交于一点.

例3

四边形

ABCD中,

???
AB

=

a



???
BC

=

b



???
CD

=

c



???
DA



d

,且

a

·b

=b

·c

= c ·d = d ·a ,试问四边形 ABCD是什么图形?

例 4 设 a 与 b 是夹角为 60°,且| a | ? | b |,是否存在
满足条件的 a , b ,使 | a + b |=2| a - b |?请说明理由.

巩固深化,反馈矫正

1.已知| a |=1,|b |= 2 ,

(1)a - b 与 a 垂直,则 a ? b 的夹角是______;

2.(已(2知3))|若若a|a=a2、/,/bb|b,的|a=夹?1b,角a?为与__?b3_之,;间则的|a夹+ b角| 为? _?__,_;那么向量

|a -4b|·|a - b | ? ____

3

a

-4

b

的模为_____;

3.设 e1、e2是两个单位向量,其夹角为600,求向量a =2e1+ e2与
b =2 e2-3e1的夹角;

4.对于两个非零向量 a,b

(1)求使|a ? tb|最小时的值,并求此时b 与a ? tb 的夹角;
(2)当 a ? tb(t ? R) 的模取最小值时,
①求t 的值;②求证:b 与a ? tb垂直.

回顾反思
通过本节学习,要求大家掌握平面向量数量积的运算规律, 掌握两个向量共线、垂直的几何判断,能利用数量积的重要性质 解决相关问题.


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