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www.x1949x.com:2019-数学:2.4.2《平面向量数量积的坐标表示、模、夹角》PPT课件(新人教A版必修4)-文档资料_图文

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新课标人教版课件系列
《高中数学》
必修4

2.4.2《平面向量数量积 的坐标表示、模、夹角》

教学目标
? 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

一、复习引入

(1) a ? b ? a ? b cos ?
2
(2)a ? a ? a 或 a ?

a ? a;

a ? b ? a ? b ? 0; cos ? ? a ? b .
a ?b

我们学过两向量的和与差可以转 化为它们相应的坐标来运算,那么怎 样用 a和b的坐标表a示 ?b呢?

二、新课学习

1、平面向量数量积的坐标表示

如图,i 是x轴上的单位向量, j 是y

轴上的单位向量,

由于a?b? a? bco?s 所以 y A(x1,y1)

B(x2,y2)

i ? i ?1 . j ? j ? 1 .

b

a
j

i ? j ? j ?i ?0 .

oi

x

下面研究怎样用

a和b的坐标表示 a?b.
设两个非零向量 a =(x1,y1), b =(x2,y2),则

a?x1i?y1 j b?x2i?y2 j,

a?b?(x1i?y1 j)?(x2i?y2 j)

2

2

?x1x2i ?x1y2i? j?x2y1i? j?y1y2 j

?x1x2 ?y1y2

故两个向量的数量积等于它们对应

坐标的乘积的和。即

y A(x1,y1)

a?b?x1x2?y1y2. B(x2,y2) a

b

j

oi

x

根据平面向量数量积的坐标表示,向 量的数量积的运算可转化为向量的坐标运 算。

2、向量的模和两点间的距离公式
2
(1)a?a?a 或 a? a?a;

(1)向量的模
2
设a ? (x, y),则 a ? x2 ? y2 ,或 a ? x2 ? y2;

(2)两点间的距离公式

设A(x1, y1)、B(x2 , y2 ),



AB ? (x1 ? x2 )2 ?(y1 ? y2 )2

3、两向量垂直和平行的坐标表示 (1)垂直 a?b? a?b?0
设a?(x1, y1),b?(x2, y2),则 a?b?x1x2 ?y1y2 ?0
(2)平行
设a?(x1, y1),b?(x2, y2),则 a//b?x1y2 ?x2y1 ?0

4、两向量夹角公式的坐标运算
设a与b的夹角为 ?(0? ?? ?180?), 则cos? ? a?b
ab
设a?(x1, y1),b?(x2, y2),且a与b夹角为 ?, (0? ?? ?180?)则co?s ? x1x2 ?y1y2 .
x12 ?y12 ? x22 ?y22 其中x12 ?y12 ?0,x22 ?y22 ?0.

三、基本技能的形成与巩固 例 1 (1已 ) a知 ?(?1,2? 3)b , ?(1,1),
求 a?b, a?b, a与 b的夹 ?. 角
a?b?1? 3, a?b?2 4?2 3?2(1? 3),
co?s? a?b ?1,?0? ???18?0,???60?.
a?b 2

(2已 ) 知 a?(2,3),b?(?2,4),

则( a?b) ( ? a?b) ?

.

法一:a ? b ? (0,7),a ? b ? (4,?1)

?(a ? b)(? a ? b)? 0? 4 ? 7 ? (?1) ? ?7.

法二:(a

?

b)(? a

?

b)?

2
a

?

2
b

22
? a ? b ? 13? 20 ? ?7

练习:课本P1191、2、3.

例2 已知A(1,2),B(2,3),C(-2,5),

试判断?ABC的形状,并给出证明.
y

证 :? 明 A? B (2? 1 ,3?2 )?(1 ,1 )C(-2,5)

A? C (?2?1,5?2)?(?3,3)

B(2,3)

? A?A B? C 1?(? 3)?1?3?0

A(1,2)

?AB?AC

x

?三角A形B是 C 直角三.角形0

练习2:以原点和A(5,2)为两 个顶点作等腰直角三角形OAB, ?B=90?,求点B的坐标.

答案:B的坐标为(3,7) y B

22

或(7,? 3)

A

22

O

x

四、逆向及综合运用

例3 (1)已知 a =(4,3),向量 b 是
垂直于 a 的单位向量,求 b .

( 2)已 a?知 1,0 b?(1,2),a且 //b,a的 求坐 .

( 3)已 a?知 (3,0)b , ?(k,5),a且 与 b的夹3角 ?,
4

求 k的.值

答案 1) b: ?(3,? ( 4)或 b?(?3,4).

55

55

( 2)( 2, 22)或 ?2, ( ?22); 3) k( ??5.

提高练习
1、已 O? A 知 (?3,1), O? B(0,5),A 且 /C O / , B BC ?A, B 则 C的 点 坐标 C(?3, 23为 9 )
2、已知A(1,2)、B(4、0)、C(8,6)、 D(5,8),则四边形ABCD的形状是矩形.
3、已知 a = (1,2),b = (-3,2), 若k a +2 b 与 2 a - 4b 平行,则k = - 1.

小结 1、理解各公式的正向及逆向运用; 2、数量积的运算转化为向量的坐 标运算; 3、掌握平行、垂直、夹角及距离 公式,形成转化技能。
作业 课本P119A组5(1),9,10,11.


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