the question is here for clarity
If three lines l_1, l_2, l_3 are given as l_1={(s,0,1)|s an element of R} and l_2={(1,t,0)|t an element of R} and l_3={(0,1,u)|u an element of R} then show that there are infinitely many lines that cross l_1,l_2 and l_3 at the same time.
I tried to separate the given equations of the lines like l_1=(0,0,1)+s(1,0,0), l_2=(1,0,0)+t(0,1,0) and l_3=(0,1,0)+u(0,0,1) but it turns out that from the above equations, the direction vectors of these three lines are perpendicular to one another and the direction vectors turned out to be the standard i, j and k. I thought the three lines would be all on the same plane and not perpendicular to one another. Is there something that I am missing? Any hint or a reference to a book with similar problems will really be appreciated. Please...