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平面向量的数量积

一、引入:

一个物体在力F 的作用下产生 的位移s,那么力F 所做的功 应当怎样计算?

F

θ

S

力做的功:W = |F|?|s|cos?,?是F与s的夹角

向量的数量积

?1.两个非零向量夹角的概念

?已知非零向量a与b,作 O A=a,O B =b,则∠AOB=θ
(0≤θ≤π)叫a与b的夹角.

说明:
(1)当θ=0时,a与b同向;
(2)当θ=π时,a与b反向;
(3)当θ=π/2时,a与b垂直,

b

a

O

ba
O

记a⊥b;

?(4)注意在两向量的夹角定义中,两向量必 须是同起点 的.范围0?≤?≤180?

平面向量数量积(内积)的定义:
已知两个非零向量a与b,它们的夹角是θ,则数量 |a||b|cos?叫a与b的数量积,记作a?b,即有
a?b =|a||b|cos?,(0≤θ≤π).
规定0与任何向量的数量积为0。 ?探究:两个向量的数量积与向量同实数积有很大区别
(1)两个向量的数量积是一个实数,不是向量,符号由 cos?的符号所决定。 (2)两个向量的数量积称为内积,写成a?b;符号“· ”在 向量运算中不是乘号,既不能省略,也不能用“×”代替. (3)在实数中,若a?0,且a?b=0,则b=0;在数量积中, 若a?0,且a?b=0,能不能推出b=0?为什么? (4)由a?b = b?c 能否推出a = c ?
(5)在实数中,有(a?b)c = a(b?c),但是(a?b)c ? a(b?c) 显然,这是因为左端是与c共线的向量,而右端是与a共
线的向量,而一般a与c不共线。

3.“投影”的概念: 定义:|b|cos?叫做向量b在a方向上的投影。
投影也是一个数量,不是向量; 当?为锐角时投影为正值; 当?为钝角时投影为负值; 当?为直角时投影为0;
当? = 0?时投影为 |b|; 当? = 180?时投影为 ?|b|。
4.向量的数量积的几何意义: 数量积a?b等于a的长度与b在a方向上投影|b|cos?的乘积。

?5.两个向量的数量积的性质:
设a、b为两个非零向量,e是与b同向的单位向量。
1? e?a = a?e =|a|cos?
2? a?b ? a?b = 0 3? 当a与b同向时,a?b = |a||b|;当a与b反向时,
a?b = ?|a||b|。 ?特例:a?a = |a|2或 | a|? a?a
a ?b
4? cos? =
| a || b |
5? |a?b| ≤ |a||b|

例1 判断正误,并简要说明理由.

①a·0=0;

②0·a=0;

③0- BA = A B ; ④|a·b|=|a||b|;
⑤若a≠0,则对任一非零b有a·b≠0;

⑥a·b=0,则a与b中至少有一个为0;

⑦对任意向量a,b,с都有(a·b)с=a?(b·с);

⑧a与b是两个单位向量,则a2=b2.

例2 已知|a|=3,|b|=6, 当①a∥b,②a⊥b, ③a与b的夹角是60°时,分别求a·b.

例3 判断下列命题的真假: 在△ABC中,若 ABBC?0 ,则△ABC是锐角三角形; 在△ABC中,若 ABBC?0 ,则△ABC是钝角三角形;
△ABC为直角三角形的充要条件是 ABBC?0
例4 试证明:若四边形ABCD满足 A B ? C D ? 0 ,且 A B B C ? 0 , 则四边形ABCD为矩形.
例5 设正三角形ABC的边长为
2 , A B ? c , B C ? a , C A ? b , 求 a b ? b c ? c a .
五、作业:习题5.6 1~6. 《优化设计》P81 强化训练 1~8.


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