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2.4.2

平面向量数量积的坐标 表示、模、夹角

回顾复习:

1.向量a与b的数量积的含义是什么?

a· b=|a||b|cosθ. 其中θ为向量a与b的夹角

2.向量的数量积具有哪些运算性质?

(1)a⊥b ? a· b=0(a≠0,b≠0); (2)a2=︱a︱2; (3)a· b=b· a; (4)(λa)· b=λ(a· b)=a·(λb); (5)(a+b)· c=a · c +b · c;

3.设a、b为两个向量,且a=(x1,y1),b=(x2,y2) ? ?? ? x ? x i ? y ? y j ?? x ? x , y ? y 1 2 1 2 1 2 1 2 a+b= ? ? ?? ? x ? x i ? y ? y j 1 2 1 2 ?? x ? x , y ? y 1 2 1 2 a-b= ?

2. 探究
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?

单位向量i, j分别与x轴,y轴方向相同 1 1 0 0 i· i =_____, j· j=______, i· j=______, j· i =_______.

已知两个非零向量a=(x1,y1),b=(x2,y2),怎样用a与 b的坐标表示a· b? ∵a=x1i+y1j, b=x2i+y2j, ∴a· b = (x1i+y1j) · (x2i+y2j) = x1x2i2+x1y2i· j+x2y1i· j+y1y2j2 = x1x2+y1y2
两个向量的数量积等于它们对应坐标的乘积的和

小试牛刀
1.a=(1,2)b=(2,3) 求a ? b

?

2.a=(1,3)求a2

3.a=(1,-2)b=(2, 1)
1. 8 2. 10

?
3.

求a ? b
0

?

向量平行和垂直的坐标表示式
设a、b为两个向量,且a=(x1,y1),b=(x2,y2),

a // b ? x y ? x y ? 0 1 2 2 1
b ? x x ? y y ? 0 a· b= 0 x a⊥a b? 1 2 1 2 1x 2+y1y2=0
思考:垂直的判定必须是非零向量才成立,为什么?

向量的长度(模)
设a =(x,y),则 |a|2=
x2 ? y 2

或|a |= _______ x2 ? y2

2 2 ? ? ? ? ? ? ? x x y y 若设A(x1,y1)、B(x2,y2),则|AB|=_____________ 2 1 2 1

例1 已知向量a=(4,3),b=(-1,2), 求: (1) a· b; (2) (a+2b)· ( a - b ); (3) |a|2-4a· b. (1) 2;(2)17;(3)17
练习:课后练习1, 2

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.

? ? ? ? ? AB ? 2 ? 1 , 3 ? 2 ? 1 , 1

y C B A

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

? ? ? AB ? AC ? 1 ? ? 3 ? 1 ? 3 ? 0
?AB? AC
∴ △ABC是直角三角形

o

x

向量的数量积是否为零,是判断相应的两条线段或直 线是否垂直的重要方法之一

变式:已知A(1,2)、B(4、0)、C(8,6)、 D(5,8),判断四边形ABCD的形状. 矩形
y D C

A

注意形-垂直与数-数量积为零的相互转化 探究:数量积坐标运算与向量夹角如何转化 呢?

o

B

x

设a、b是两个非零向量,其夹角为θ , 若a=(x1,y1),b=(x2,y2),那么cosθ 如何用坐标表示?

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

例3.已知 a ? 1 , 3 , b? 3 ? 1 , 3 ? 1 ,
分析:为求a与b夹角,需先求a· b及|a||b|,再结合夹 角θ的范围确定其值. 0≤θ≤π

? ? ?

?

则 a 与 b 的夹角是多少?

解: a ? ? ? ? ? 1 , 3 , b ? 3 ? 1 , 3 ? 1

a ? b ? 3 ? 1 ? 3 3 ? 1 ? 4 知三角形函 a ?2 , b ?2 2 数值求角时, 记a与b的夹角为θ 应注重角的 a ?b 2 cos ?? ? 范围的确定 ab 2
又0≤θ≤π
? ?? ? 4

? ?

? ? ? ? 3 ? ? ? ? 变式练习:已知 a ? 3 , 0 , b ? k , 5 , 且 a , b 的夹角 ,求 k 的 4

?

K= - 5

达标练习: 1.已知

( ) a ? 1 ,b ?2 且 ( a ? b ) 与 a 垂直,则 a 与 b 的夹角是 ? A. 60 ? B. 30 ? C. 135 D. 45 ? ? 2.已知 a =2 , 的夹角是 ,那么向量 a - b 的模为( ) b =1,a与 b 3 A.2 B. 2 3 C.6 D.12

? ? ? ? 3 . a ? 2 , 1 , b ? 1 , 3 , 求 a ? b 及 a 与 b 的夹角 ? ? ? ? 4 . 已知 a ? 2 , 1 , b ? ? , 3 , 且 a ? b , 则 ? ?
参考答案:1.D 2.B 3. 5

45

?

4.

3 2

5. a=(-4,7), b =(5,2),则

a? b =

,

a

=

(2 -3b)? (a +2b)= a
?

6.已知点A(1,2),B(4,-1),问在 y 轴上找点 C ,使∠ABC=90 由;若能,求 C 坐标

,若不能,说明理

参考答案:5. -6

65

-50

6.C(0,-5)

小结作业

1.a∥b ? x y ? x y ? 0 1 2 2 1 a ⊥b . ? x x ? y y 0 1 2 1 2? 2.若非零向量a 与b的夹角为锐角(钝 角),则a· b>0(<0),反之不成立.
3.向量的坐标运算沟通了向量与解析几 何的内在联系,解析几何中与角度、距 离、平行、垂直有关的问题,可以考虑 用向量方法来解决.

作业:≤非常学案≥第51页1-8题


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