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www.6680993.com:2019年2.4.2平面向量数量积的坐标表示、模、夹角(1)精品教育.ppt_图文

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1、 a ? b ?| a || b | cos?

2设 (、1)a?a数?、?量b?b积是 ? ?的非性零a? 质向 ? b?:量? 0

(2)当a?与b?同向时,

a?

?

? b

?

a?

? b

当a?与b?反向时, a?

? ?b

?

?

a?

? b

特别地,a ? a ? a 2 或 a ? a ? a ? a 2

(3) a ? b ? a b

一、平面向量数量积的坐标表示:

i ? i ?1

y

j ? j ?1

b

a

i ? j ?0

j

oi

x

设:a ? (x1, y1),b ? (x2, y2)

则:a ? x1i ? y1 j, b ? x2i ? y2 j

一、平面向量数量积的坐标表示:

? ? a ? ? x1, y1 ? ,b ? ? x2 , y2 ? a ? 0,b ? 0

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

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

2

2

? x1x2i ? x1y2i ? j ? x2 y1i ? j ? y1y2 j

2

2

i ? 1, i ? j ? 0, j ? 1

?a ?b ? x1x2 ? y1y2

二、向量的模和两点间距离公式:
?1?向量的模(长度公式):设a ? (x, y),
则 ? a ?2 ? a ? a ? x2 ? y2 , 或 ? a ?? x2 ? y2
?2?设A? x1, y1?、B?x2, y2 ?, 则AB ? ? x2 ? x1, y2 ? y1 ?
? AB ?? ? x2 ? x1 ?2 ? ? y2 ? y1 ?2

三、向量夹角公式的坐标表示:
设a ? ?x1, y1 ?,b ? ?x2 , y2 ?, a与b夹角为?,?0 ? ? ? ? ?
cos? ? a ? b ? x1x2 ? y1 y2
| a || b | x12 ? y12 . x22 ? y22

例1:?1?已知a ? (3, ?1),b ? (1, ?2),
求a ?b,| a | ? | b |,a与b的夹角?.
解? a ?b ? x1x2 ? y1y2 ? 5
| a | ? | b |? x12 ? y12 . x22 ? y22 ? 5 2
cos? ? a ?b ? 5 ? 2
| a || b | 5 2 2
?? ? ?
4

?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

?| a |2 ? | b |2 ? 13 ? 20 ? ?7

四、向量垂直和平行的坐标表示:
a ? ? x1, y1 ? ,b ? ? x2, y2 ?(a ? 0,b ? 0)
(1)垂直: a ? b ? a ?b ? 0
? x1x2 ? y1 y2 ? 0
(2)平行: a / /b ? b ? ?a
? x1 y2 ? x2 y1

例2:已知A(1, 2),B(2, 3),C(?2, 5), 求证:△ABC是直角三角形
证明: ? AB ? ?1,1? AC ? ??3,3? BC ? ?? 4,2?
? AB ? AC ? ?3 ? 3 ? 0
即AB⊥AC, △ABC是直角三角形.
想一想: 还有其他解法吗?
AB ? 2, AC ? 3 2, BC ? 2 5

例3:已知a ? (4,3),求与a垂直的单位向量b

解:设所求向量为(x, y), 则

?4x ? 3y ? 0

? ?

x2

?

y2

?1

?

? ? ? ?

yx???534或????xy???453

? 5? 5

?b ? (3 ,? 4)或b ? (? 3 , 4)

55

55

例4:已知a =(1, 0),b =(2, 1),当k为何实数 时,向量k a- b与 a+3b(1)平行;(2)垂直
解:k a- b =(k-2, -1) a +3 b=(7, 3)
(1)由向量平行条件得3(k-2)=-7 所以k= ? 1 3
(2)由向量垂直条件得7(k-2)-3=0 所以k= 17
7

练习
B

C

D

23 3

1

m=-2
?
6

A

小结
1.a ?b ? x1x2 ? y1y2 2. AB ? ? x2 ? x1 ?2 ? ? y2 ? y1 ?2
3.a ? b ? x1x2 ? y1y2 ? 0
4.a / /b ? x1y2 ? x2 y1 5.cos? ? x1x2 ? y1y2
x12 ? y12 . x22 ? y22


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