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 question ask to calculate and form the parametric representation for the surface and the plane through the origin that contains the vector i-j and j-k, in that fact, base on my formulation, the equation of the surface in parametric form is S:(U,V,-U-V). I hope this would help 

Answer 2

Parametric representation for the surface in term of (x.y,z) are, x = u , y = -u + v and z = -v

Parametric representation based problem:

We know that;

If it is  through the origin, <0,0,0>

The vectors = i-j , <1, -1, 0 >

The vectors = j-k, <0,  1, -1>

By solving equation for each component;

r(u,v) = <0,0,0> + u<1,-1,0> + v<0,1,-1>

x = 0 + u×1 + v×0

x = 0 + u + 0

x = 0

y = 0 + -1×u + 1×v

y = 0 -u + v

y = -u + v

z = 0 + u×0 + -1×v

z = 0 + 0 - v

z = -v

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