Question

Find a parametric representation for the surface. (Enter your answer as a comma-separated list of equations. Let x, y, and z be in terms of u and/or v.)
The plane through the origin that contains the vectors i − j and j − k.

Answer 1

the given vectors are A=<1, -1, 0> and B=<0, 1, -1>
a vector normal to both A and B is given by cross product N= A X B
N=<1, -1, 0> X <0, 1, -1> = <1, 1, 1>

we know that equation of a plane is in the form ax+by+cz=d and N=<a, b, c>
substitute the components of N,
we get the plane x + y +z =d

we know that (0,0,0) is a point on the plane, substitute for x,y and z in the above equation to get d=0
thus x+y+z=0 is the required plane.

let x=u and y=v
then z=-u-v

thus equation of the surface in parametric form
s: (u, v, -u-v)