...The questions asks to find the Cartesian equation of the lines r=<1,1,0>+t<1,-2,2> and r=<2,0,2>+t<-1,1,0>
Just trying to work out how to solve. Is it okay to take the point vector and direction vector of each line equation above and do a cross product on them to get a vector normal ie <1,1,0> x <1,-1,2> = <2,-2,-2>= n Where came these vectors from?
than a dot product <2,-2,-2> . <x-1, y-1, z-0>=0 This is the equation of a plane passing through (1,1,0)
to get the Cartesian equation of the first line
and similarly do the same for the second line
Than elimate x from first equation, to get expression for y etc
Then you'll get maybe the intersection of 2 planes which could be a line, but not the equation of the given lines.