在△ABC中,已知D是AB邊上一點,若
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_ST/0.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_ST/1.png)
,則λ
1-λ
2=
.
【答案】
分析:由題意知,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/0.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/1.png)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/2.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/3.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/4.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/5.png)
,待定系數(shù)法求出λ
1和λ
2的值.
解答:解:∵
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/6.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/7.png)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/8.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/9.png)
(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/10.png)
-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/11.png)
)=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/12.png)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/13.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/14.png)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/15.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/16.png)
,
又∵
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/17.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/18.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/19.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/20.png)
+(-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/21.png)
)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/22.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/23.png)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/24.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/25.png)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/26.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/27.png)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/28.png)
,
∴λ
1=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/29.png)
,λ
2=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/30.png)
;
∴λ
1-λ
2=![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180928768370418/SYS201310241809287683704010_DA/31.png)
.
點評:本題考查向量運算法則、向量的共線定理,待定系數(shù)法.
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