9.已知在四面体ABCD中,(AB) ⃗=a-2c,(CD) ⃗=5a+6b-8c,棱AC,BD的中点分别为E,F,则(EF) ⃗=　　　　　　.

解析:如图所示,取BC的中点G,连接EG,FG,

　　则(EF) ⃗=(GF) ⃗-(GE) ⃗=1/2 (CD) ⃗-1/2 (BA) ⃗=1/2 (CD) ⃗+1/2 (AB) ⃗

=1/2(5a+6b-8c)+1/2(a-2c)=3a+3b-5c.

答案:3a+3b-5c

10.

如图所示,已知平行六面体ABCD-A'B'C'D',点E在AC'上,且AE∶EC'=1∶2,点F,G分别是B'D'和BD'的中点,求下列各式中x,y,z的值.

(1)(AE) ⃗=x(AA"'" ) ⃗+y(AB) ⃗+z(AD) ⃗;

(2)(BF) ⃗=x(BB"'" ) ⃗+y(BA) ⃗+z(BC) ⃗;

(3)(GF) ⃗=x(BB"'" ) ⃗+y(BA) ⃗+z(BC) ⃗.

解(1)∵AE∶EC'=1∶2,

∴(AE) ⃗=1/3 (AC"'" ) ⃗=1/3((AB) ⃗+(BC) ⃗+(CC"'" ) ⃗)

=1/3((AB) ⃗+(AD) ⃗+(AA"'" ) ⃗),

∴(AE) ⃗=1/3 (AA"'" ) ⃗+1/3 (AB) ⃗+1/3 (AD) ⃗,∴x=y=z=1/3.

(2)∵F是B'D'的中点,

∴(BF) ⃗=1/2((BB"'" ) ⃗+(BD"'" ) ⃗)

=1/2((BB"'" ) ⃗+(BA) ⃗+(AA"'" ) ⃗+(A"'" D"'" ) ⃗)

=1/2(2(BB"'" ) ⃗+(BA) ⃗+(BC) ⃗)=(BB"'" ) ⃗+1/2 (BA) ⃗+1/2 (BC) ⃗,

∴x=1,y=z=1/2.