The parametrization of the segment line limited by O(0,0,0) and I(1,1,0) is r(t) = ti + tj, for t between 0 and 1
Take any point M (x,y,z) inside the segment line. OM = r is parallel to OI(1,1,0). This means that exists t such as OM = t OI.
On x axis this gives : x = t
On y axis : y = t
On z axis : z = 0
Therefore M(t,t,0) or OM = ti + tj
I hope my explanation will be of help for you.