∵向量OP·OQ=0,
∴向量OP⊥OQ,
设P(x1,y1),Q(x2,y2),
则x1x2+y1y2=0,
设直线方程为:y=k(x-1),.(1)
y1=k(x1-1),
y2=k(x2-1),
y1y2=k^2[x1x2-(x1+x2)+1].
x1x2+y1y2=x1x2(1+k^2)-k^2(x1+x2)+k^2=0,.(2)
(1)式代入椭圆方程,
x^2/3+k^2(x-1)^2=1,
(1+3k^2)x^2-6k^2x+3k^2-3=0,
根据韦达定理,
x1+x2=6k^2/(1+3k^2),.(3)
x1x2=3(k^2-1)/(1+3k^2),.(4)
由(3)式、(4)式代入(2)式,
[3(k^2-1)/(1+3k^2]*(1+k^2)-k^2(6k^2/(1+3k^2)+k^2=0,
3(k^4-1)/(1+3k^2)-6k^4/(1+3k^2)+k^2=0,
k^2=3,
∴k=±√3,
∴直线l的方程为:y=±√3(x-1).
可以作与椭圆有二交点PQ直线l,且向量OP·OQ=0.