已知长方体ABCD-A'B'C'D',点E,F分别是上底面A'B'C'D'和平面CC'D'D的中心,求下列各题中的xyz的值（1）AC'=xAB+yBC+zCC'(2)AE=xAB+yBC+zCC'(3)AF=xBA+yBC+zCC'其中的大写字母都为向量

Question

已知长方体ABCD-A'B'C'D',点E,F分别是上底面A'B'C'D'和平面CC'D'D的中心,求下列各题中的xyz的值
（1）AC'=xAB+yBC+zCC'
(2)AE=xAB+yBC+zCC'
(3)AF=xBA+yBC+zCC'
其中的大写字母都为向量

Answer 1

（1）AC'=xAB+yBC+zCC'
AC'=AB+BC+CC'
∴x=y=z=1
(2)AE=xAB+yBC+zCC'
AE=AA'+A'E
=CC'+1/2(A'C')
=CC'+1/2AC
=1/2AB+1/2BC+CC'
x=y=1/2,z=1
(3)AF=xBA+yBC+zCC'
AF=AD'+D'F
=BC'+1/2D'C
=BC+CC'+1/2(-CC'+AB)
=1/2AB+BC+1/2CC'
∴x=z=1/2,y=1