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haoka8.com:人教版高中数学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)_图文

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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的坐标表示a ? 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 ? b cos? 所以 y A(x1,y1)
B(x2,y2)
i ? i ?1 . j ? j ?1 . b j a

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

? x1x2 i ? x1 y2 i ? j ? x2 y1i ? j ? y1y2 j

? x1x2 ? y1 y2

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 ? y1 y2. B(x2,y2) a
bj
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 ? y1 y2 ? 0
(2)平行
设a ?(x1, y1), b ? (x2 , y2 ), 则 a//b ? x1 y2 ? x2 y1 ? 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cos? ? x1x2 ? y1 y2 .
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)已知a ? (?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),
cos? ? a ? b ? 1 ,? 0? ? ? ? 180? ,?? ? 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)?

a

2

?

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:?AB ? (2 ?1,3 ? 2) ? (1,1)C(-2,5)

AC ? (?2 ?1,5 ? 2) ? (?3,3)

B(2,3)

?AB? AC ? 1? (?3) ?1? 3 ? 0

A(1,2)

?AB ? AC

x

?三角形ABC 是直角三角形.

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

A

或(7 ,? 3)

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)已知 a ? 10 ,b ? (1,2),且a // b,求a的坐标.

(3)已知a ? (3,0),b ? (k,5),且a与b的夹角为3? ,
4

求k的值.

答案:(1)b ? (3 ,? 4)或b ? (? 3 , 4).

55

55

(2)( 2,2 2)或(? 2,? 2 2);(3)k ? ?5.

提高练习
1、已知OA ? (?3,1),OB ? (0,5),且AC // OB, BC ? AB,则点C的坐标为 C(?3, 29)
3
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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