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yufei999:2.4.3平面向量数量积的坐标表示-PPT课件_图文

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一、平面向量数量积的坐标表示:
a ? x , y , b ? x , y a , b 非 零 向 量 ? ? ? ? 1 1 2 2

?

?

? a ? x i ? y j , b ? x i ? y j 1 1 2 2

? a ? b ? ( x i ? y j )( x i ? y j ) 1 1 2 2

? x x i ? x y i ? j ? x y i ? j ? y y j 1 2 1 2 2 1 1 2
? i ? 1 , i ? j? 0 ,j ? 1
2 2

2

2

? a ? b ? x x y y 1 2? 1 2

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 2 2 2 x? y , 或 a ? x? y
2

? ? 2 两点间的距离公式: ? ? ? ? ? ? 设 A x ,y 、 B x ,y ,则 AB ? x x ,y y 1 1 2 2 2? 1 2? 1

?? ? ? A B ?? x ? x y ? y 2 2
2 1

2 1

三、向量垂直和平行的坐标表示:

? ? ? ? a ? x , y , b ? x , y 1 1 2 2
(1)垂直:

a ? b ? a ? b ? 0 ? x x ? y y ? 0 1 2 1 2
(2)平行:

a // b ? x y ? x y ? 0 1 2 2 1

四、向量夹角公式的坐标表示:

设 a??x ,y ,b??x , 1 1? 2, y 2?

?0????? a 与 b 夹角为 ? ,
a ? b x x y y 1 2? 1 2 cos ?? ? 2 2 2 2 x y y a .b 1? 1. x 2? 2

例 1 :?? 1 已知 a? ( 3 ,? 1 ), b? ( 1 ,? 2 ), 求 a ? b , a? b , a 与 b 的夹角 ? . a ? b? 5

? ? ? ? ? ? 2 已知 a? 2 , 3 ,b? ? 2 ,4 ,
则 a ? b?a ? b?

? ?? ?
? ?? ?

a ? b?5 2

.

? ? ? ? a ? b ? 0 , 7 , a ? b ? 4 , ? 1

? ? ? 4

? ? ? a ? b ? a ? b ? 0 ? 4 ? 7 ? ? 1 ? ? 7

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.

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

?4 x ? 3 y ? 0 ? 2 2 ? x ? y ?1

3 ? 3 ? ? x ? 5 ?x ? ? 5 ?? 或? 4 4 ?y ? ? ? y? 5 ? 5 ?

3 4 34 ? b ? (, ?) 或 b ? ( ?, ) 5 5 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 +3 b (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=0

1 所以k= ? 3 (2)由向量垂直条件得7(k-2)-3=0
1 7 所以k= 7

提高练习:
29 C (?3, ) BC ? AB ,则点 C 的坐标为 3

1 、已知 OA ? ( ? 3 , 1 ) , OB ? ( 0 , 5 ) ,且 AC // OB ,

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 .

a = (1,2),b = (-3,2), 若k a +2 b 与 2 a - 4 b 平行,则k = - 1
3、已知

.

【小结】
(1)掌握平面向量数量积的坐标表示,
即两个向量的数量积等于它们对应坐

标的乘积之和; (2)要学会运用平面向量数量积的坐标
表示解决有关长度、角度及垂直问题.

P


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