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49491.com:【数学】2.4.2平面向量数量积的坐标表示、模、夹角-PPT精选文档_图文

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第二章 平面向量
2.4 平面向量的数量积

2.4.2 平面向量数量积的坐标 表示、模、夹角

问题提出

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; (6)︱a· b︱≤︱a︱︱b︱.

3.平面向量的表示方法有几何法和坐标 法,向量的表示形式不同,对其运算的 表示方式也会改变.向量的坐标表示,对 向量的加、减、数乘运算带来了很大的 方便.若已知向量a与b的坐标,则其数量 积是唯一确定的,因此,如何用坐标表 示向量的数量积就成为我们需要研究的 课题.

探究(一):平面向量数量积的坐标表示

思考1:设i、j是分别与x轴、y轴同向的 两个单位向量,若两个非零向量a=(x1, y1),b=(x2,y2),则向量a与b用i、j分别 如何表示?
a = x 1 i+ y 1 j, b= x 2 i+ y 2 j. 思考2:对于上述向量i、j,则i2,j2,i· j 分别等于什么? i2=1,j2=1,i· j=0.

思考3:根据数量积的运算性质,a· b等 于什么? a· b= x 1 x 2 + y 1 y 2 思考4:若a=(x1,y1),b=(x2,y2),则 a· b=x1x2+y1y2,这就是平面向量数量 积的坐标表示.你能用文字描述这一结论 吗?

两个向量的数量积等于它们对应坐标的 乘积的和.

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思考5:如何利用数量积的坐标表示证明 (a+b)· c= a · c+ b· c?

探究(二):向量的模和夹角的坐标表示

思考1:设向量a=(x,y),利用数量积 的坐标表示,︱a︱等于什么? 2 2 ︱a︱= x + y 思考2:如果表示向量a的有向线段的起点 和终点的坐标分别为(x1,y1), (x2,y2), 那么向量a的坐标如何表示?︱a︱等于什 么? a=(x2-x1,y2-y1);
x x )? ( y y ) ︱a︱= ( 2? 1 2? 1
2 2

思考3:设向量a=(x1,y1),b=(x2,y2), 若a⊥b,则x1,y1,x2,y2之间的关系如何 反之成立吗? a⊥ b ? x 1x 2 +y 1y 2=0 .
?

思考4: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

理论迁移

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)-3.

例2 已知点A(1,2),B(2,3), C(-2,5),试判断△ABC的形状,并给 出证明. △ABC是直角三角形
例3 已知向量a=(5,-7),b= (-6,-4),求向量a 与b的夹角θ (精确到1°).

cosθ ≈-0.03,θ ≈92°.

例4 已知向量a=(λ ,-2),b= (-3,5),若向量a 与b的夹角为钝角, 求λ 的取值范围.

1 06 6 ( , )U ( ,+? ) 3 5 5
例5 已知b=(1,1),a· b= 3 , |a-b|=2,求|a|.

2 2

小结作业

1.a∥b ? x y ? x y ? 0 1 2 2 1 a ⊥b ? 二者有着 x x ? y y 0 1 2 1 2? 本质区别. 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.


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