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wjff2001:最新2019-242平面向量数量积的坐标表示、模、夹角1-PPT课件_图文

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2.4.2 平面向量数量积的 坐标表示、模、夹角(1)

一:平面两向量数量积的坐标表示
已知两个非零向量 a?(x1,y1) , b?(x2,y2), 试用 a 和 b 的坐标表示 a ? b.

? a?x1i?y1j,b?x2i?y2j

? a?b?(x1i?y1j)x (2i?y2j)

2

2

? x 1 x 2 i? x 1 y 2 i?j? x 2 y 1 i?j? y 1 y 2j?x1x2?y1y2

?a?b?x1x2?y1y2

两个向量的数量积等于它们对应坐标的乘积的和

二: 公式与结论
1 向量模的坐标表示 a?(x,y)? |a|? x2?y2
2 两点间的距离公式 已知两点 A(x1,y1) 、B(x2,y2) 则 A? B(x2?x1,y2?y1)? |A|? B(x1?x2)2?(y1?y2)2
3 已知 a?(x1,y1), b?(x2,y2)
a?b? x1x2?y1y2?0 a/b /? x1y2?x2y1?0
两向量夹角的余弦( 0????) cos? ? a ?b ? x1x2?y1y2
| a | ? | b | x12?y12 x22?y22

三:应用举例 例1 已知A(1, 2),B(2, 3),C(?2, 5),
试判断△ABC的形状,并给出证明.
例2 设? a = (5, ?7),b?= (?6, ?4),
?? ? ?
求a ·b及a、b间的夹角θ(精确到1o)

例3 已知a=(1,3),
b=( 3+1, 3-1), 求 a 与 b 的夹角;
例4 已知a = (3, ?1),b = (1, 2),求 满足x ? a = 9与 x ? b = ?4的向量 x.

例5 以原点和A(5, 2)为顶点作等腰直角△OAB,使 ?B = 90?,求点B和向量 AB 的坐标.

解:设B点坐标(x, y),

Y

则 OB = (x, y),AB = (x?5, y?2)

B



?? O B ?

?

AB

??| O B |? | A B |

?x(x?5)?y(y?2)?0

O

得 ? ?x2?y2?(x?5)2?(y?2)2

A X
B

????1x20x??y42y??5x2?92y?0????????xy11???7223或?????xy22??2372

∴B点坐标

(7 ,? 2

3)或
2

(3 2

,7) 2

; AB

=(? 3 ,? 7) 或 (? 7 , 3 )

22

22

例6 在△ABC中,AB =(2, 3),AC =(1, k),

且△ABC的一个内角为直角,求k值.

解:当A = 90?时,AB? AC = 0,∴2×1 +3×k = 0

∴k = ?

3 2

当B = 90?时, AB ? BC= 0,BC = A C ?AB = (?1, k?3)

∴2×(?1) +3×(k?3) = 0 ∴k = 11
3
当C = 90?时, AC ? BC = 0,

∴?1 + k(k?3) = 0 ∴k = 3 ? 13

2

综上所述 k??3 或11或3? 13

23

2

作业: (1)书P108 中 A组9,10,11;
B组1,2,3,4
(2)AB本相关内容


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