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wwwaomen3322com:高中数学 第二章 平面向量 2.4 向量的数量积(3)教案 苏教版必修4

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高中数学 必修4

创设情景,揭示课题:
1.两平面向量垂直的条件;
2.两向量共线的坐标表示;
3. x 轴上单位向量 i ,y 轴上单位向量 j ,则:i ?i ? 1,j ? j ? 1,
i? j ? j?i ? 0.
提出问题:向量的数量积能否用坐标表示?

学生活动:

提出问题:设
? j

? a ? (x1, y1),b ? (x2 , y2 ) ,设 i
是 y 轴上的单位向量,试用

是 ?

x

?轴上的单? 位向? 量,

i ,j 表示 a 和 b .

a ? x1i ? y1 j, b ? x2 i ? y2 j

建构数学:

提出问题:能否用

? a



? b 的坐标表示

a?b



1.向量数量积的坐标表示:

2

2

a ?b ? (x1i ? y1 j)(x2i ? y2 j) ? x1x2i ? x1y2i ? j ? y1x2 j ?i ? y1y2 j



? i

? ?i

? 1,?j ?

? j

? ? 1 ,i

?

? j

?

?? j ?i

?

0

从而得向量数量积的坐标表示公式:a ? b ? x1x2 ? y1 y2
这就是说:两个向量的数量积等于它们对应坐标的乘积的和.
a ? b ? x1x2 ? y1 y2

建构数学:

2.长度、夹角、垂直的坐标表示:

(1)长度:设

a ? (x, y),则

2
a

2
?a

?

x2

?

y2

?

a

?

x2 ? y2 ;

(2)两点间的距离公式:若 A(x1, y1), B(x2, y2 ) ,则

???

| AB |? (x2 ? x1 ) 2 ? ( y2 ? y1 ) 2



(3)夹角:cos? ? a ?b ?
|a|?|b|

x1x2 ? y1 y2 x12 ? y12 ? x22 ? y22

( 0 ? ? ? ?);

(4)设 a

? (x1, y1), b

? (x2 , y2 )

,则

? a

?
? b ? x1x2

?

y1 y2

?0



例题讲解:
例1 设 a ? (5, ?7),b ? (?6, ?4),求 a ? b .

例题讲解:
例2 已知 a ? (x1, y1),b ? (x2 , y2 ) ,求(3 a - b)·( a -2 b ).

例题讲解:
例3 在△ABC中,设 AB ? (2,3) ,AC ? (1, k) ,且△ABC是直角三角形,
求k的值.
变式:已知 A(1, 2), B(2,3),C(?2,5) ,求证 ?ABC 是直角三角形.
说明:两个向量的数量积是否为零,是判断相应的两条直线 是否垂直的重要方法之一.

例题讲解:

例4 如图,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.

y

x

巩固深化,反馈矫正:

1.已知 a ? (cos?,sin? ), b ? (cos ? ,sin ? ) ,0 ? ? ? ? ? ? ,

(1)求证:(a ? b) ? (a ? b)

(2)若 ka ? b 与 a ? kb 的模相等,且 k ? 0 ,求? ?? 的值.

2.已知

a

?
=(?3,4),b

=(4,3),求

x,

y

的值使(

x

a

+

?
yb

)⊥

a



且|x a + yb|=1.

分析:这里两个条件互相制约,注意体现方程组思想.

归纳整理,整体认识:
1.平面向量数量积的坐标公式;向量垂直的坐标表示的条件, 复习向量平行的坐标表示的条件;
2.向量长度(模)的公式及两点间的距离公式和夹角公式.


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