Find a parametric representation for the surface. The plane through the origin that contains the vectors i - j and j - k

parametric-surface

#1

Find a parametric representation for the surface. The plane through the origin that contains the vectors i - j and j - k

Answer:

It’s through the origin, <0,0,0>, and the vectors i-j, <1,-1,0>,
and j-k, <0,1,-1>.

Setup the following equation and solve for each component:
r(u,v) = <0,0,0> + u<1,-1,0> + v<0,1,-1>

x= 0 + u1 + v0
y= 0 + -1u + 1v
z= 0 + u0 + -1v

Simply to get the following answer:

x=u,y=-u+v,z=-v