已知向量e1,e2是平面内两个不共线的向量,向量AB=2e1+e2,向量BE=-e1+xe2,EC=-2e1+e2,且AEC三点共线.求x2

Question

已知向量e1,e2是平面内两个不共线的向量,向量AB=2e1+e2,向量BE=-e1+xe2,EC=-2e1+e2,且AEC三点共线.求x2

Answer 1

A,E,C三点共线
let
|AE|/|EC|=k
BE = (BA+kBC)/(1+k)
=(-AB+k(BE+EC)/(1+k)
BE=(-AB+kEC)
-e1+xe2 = (-2e1-e2+ k(-2e1+e2) )
(1+2k)e1+(x+1+k)e2=0
=>
1+2k= 0 and (x+1+k)=0
k=-1/2 and (x+1+k)=0
=> x+1-1/2=0
x=-1/2