已知a、b是兩個(gè)非零向量,當(dāng)a+tb(t∈R)的模取最小值時(shí),
(1)求t的值;
(2)求證:b⊥(a+tb).
【答案】
分析:(1)設(shè)出兩個(gè)向量的夾角,表示出兩個(gè)向量的模長(zhǎng),對(duì)于模長(zhǎng)形式,通常兩邊平方,得到與已知條件有關(guān)的運(yùn)算,整理成平方形式,當(dāng)?shù)讛?shù)為零時(shí),結(jié)果最�。�
(2)本題要證明兩個(gè)向量垂直,這種問題一般通過向量的數(shù)量積為零來證明,求兩個(gè)向量數(shù)量積,根據(jù)上一問做出的結(jié)果,代入數(shù)量積的式子,合并同類項(xiàng),得到數(shù)量積為零.得到垂直.
解答:(1)解:設(shè)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/0.png)
與
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/1.png)
的夾角為θ,
∵|
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/2.png)
+t
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/3.png)
|
2=(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/4.png)
+t
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/5.png)
)
2=|
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/6.png)
|
2+t
2|
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/7.png)
|
2+2
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/8.png)
•(t
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/9.png)
)=|
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/10.png)
|
2+t
2|
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/11.png)
|
2+2t|
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/12.png)
||
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/13.png)
|cosθ
=|
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/14.png)
|
2(t+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/15.png)
cosθ)
2+|
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/16.png)
|
2sin
2θ,
∴當(dāng)t=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/17.png)
cosθ=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/18.png)
=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/19.png)
時(shí),|
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/20.png)
+t
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/21.png)
|有最小值.
(2)證明:∵
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/22.png)
•(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/23.png)
+t
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/24.png)
)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/25.png)
•(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/26.png)
-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/27.png)
•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/28.png)
)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/29.png)
•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/30.png)
-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/31.png)
•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/32.png)
=0,
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/33.png)
⊥(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/34.png)
t
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023214336028003902/SYS201310232143360280039009_DA/35.png)
).
點(diǎn)評(píng):啟發(fā)學(xué)生在理解數(shù)量積的運(yùn)算特點(diǎn)的基礎(chǔ)上,逐步把握數(shù)量積的運(yùn)算律,引導(dǎo)學(xué)生注意數(shù)量積性質(zhì)的相關(guān)問題的特點(diǎn),以熟練地應(yīng)用數(shù)量積的性質(zhì).?