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51平面向量的概念及线性运算

5.1

1









____________ ______(
____) ____


________


____ a |aa|

________

______________


0 ____



________



________

0 0

2.





()

(1)
ab ________. (2) (a b) c

__________.



a b b
a b

______

aba( b)

(1)|a|________

(2) >0 a (a)____

a

a ____ <0 a a

( )a _____ (a b)

____ 0 a ______

____

3. a ba b a
[ ] 1
2

1()OP QP MS MQ________ 2 ABCD E DC AB aADbBE ____________.

3

________
4 D ABC BC P PA BP CP0AP PD

________

5 O ABC D BC 2OAOBOC0( )

A.AO OD

B.AO 2OD

C.AO 3OD

D2AO OD



1
|a||b| ab ABCD AB DC ABCD

abbc acab |a||b| ab.

________



(1) a b |a|>|b| a>b

(2)|a||b| a b

(3)|a||b| a b ab

(4)

(5) a b a b
(6)AB CD ABCD

(7)

(8)

2 ABC DE BCAC
G BE GB2GEAB aAC b a b ADAG.
ABC EF ACAB
BE CF G AB aAC b a b AG.
3 a b
(1)AB abBC 2a8bCD3(ab)ABD
(2) k kab akb
ABC AC N AN13AC AB M AM13AB BN P
NP12BN CM QMQCMAP QA

11.
(13 )ABO OC14OAOD12OB AD BC MOAaOBb. a b OM.



1()


2AB CD AB CD ABCDAB BC ABC



1



2

:

1





a0 ( )

ab a b

()

A1

B2

C3

D4

2 P ABC BC BA 2BP

A.PA PB 0

B.PCPA 0

C.PB PC0

D.PA PB PC 0

()

3 ab ckab (kR)dab. cd

()

Ak1 c d

Bk1 c d

Ck1 c d

Dk1 c d 4 ab a b t atb13(ab)





1 0 1



2 (1)ba (2)a(bc) (1)|||a| (2) 0 a

aa ab




1. OS

2.b12a

3.

4.2

5.A

1

1 (1)(2)

(3)(4)(5)
(6)AB CD

AB CD ABCD.(7)(8)


[ 2 AD12(AB AC )12a12bAGAB BGAB 23BE AB 13(BA BC ) 23AB 13(AC AB )13AB 13AC 13a13b. 2 AG AB BGAB BE AB 2(BA BC )12AB 2(AC AB )(1)AB 2AC (1)a2b. AGACCGAC mCF AC m2(CA CB )(1m)ACm2 AB m2 a(1m)b

1m2 1m2

m23AG13a13b.

3 (1) AB abBC 2a8bCD3(ab) BDBC CD2a8b3(ab)2a8b3a3b5(ab)5AB . AB BD BABD

(2) kab akb kab(akb)

kabakb.(k)a(k1)b.ab

kk10k210.k1.

3

1 2

1C 2.B 3.D4 OAaOBtbOC13(ab)

ACOCOA23a13bAB OBOAtba. ABC AC

AB .23a13btba.

23 13t

23 t12.

t12

OMmanbAMOMOA manba(m1)anb.


AD

ODOA 12OB OA a12b.

AMD AMAD tAMtAD

(m1)anbta12b. (m1)anbta12tb.

m1t

n2t

t m12n

m2n1.CMOMOCmanb14am14anb CB OBOCb14a14ab.CMB CMCB t1CMt1CB m14anbt114ab

m4114t1
nt1

t1 4mn1. m17n37OM17a37b.