How do you find a parametric representation for a specific surface?

Question

I trying to find a parametric representation of the plane which goes through the origin and contains the vectors $i-j$ and $j-k$. I found the cross product for these vectors and found that the formula for the plane is $x+y+z=0$. Now I assigned $x=u$ and $y=v$ and from that determined that $z=-u-v$.

The answer given in my book is $x=u$, $y=v-u$, and $z=-v$. Is this essentially the same answer as mine?

Answer 1

There is no "absolute correct" answer, you can choose whatever you want to simplify your question.

For example, if you want to do x=u+v,y=u-v,z=-2u, it's absolutely ok as long as it can simplify your problem.

Parametric form is just a way to simplify your integral/area.

Answer 2

Yes, it is the same, from the book's answer, rewrite: $x+y+z= u+v-u-v=0$