Does the equation $$(x,y,z) = (2,1,1) +s(-1,-1,-1) + t(2,-2,-2)$$ represent a line or plane?
I claimed it is a plane, as the two direction vectors are not multiples and thus for any values of $s$ and $t$, we can get infinite points on the plane. Is this true?
The idea is close to being correct. The correct way of looking at the two direction vectors is to ask whether they are linearly independent. In case of two vectors this turns out to be the same as not being multiples of each other. So yes, this is a plane. (For three dimensional subspaces of, say, a four dimensional space this would be a bit more complicated). (Edit: And the part with 'infinite number of points' is also not really helping, a line also contains an infinite number of points. This is why I wrote 'close' to being correct...)