Find a parametric representation for the surface Enter your

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.

Please include steps taken to answer the problem. This way i can learn since i have aother questions like this. Thank you

Solution

We have vectors i - j and j - k, so their cross-product will be normal to the plane.

l i j k l
l1 -1 0 l
l 0 1 -1 l

= i + j + k. So since this vector is normal to the plane in question, and this plane

passes through the origin, the equation of the plane is x + y + z = 0.

Edit: I assume that for u and v you mean u = i - j and v = j - k.

Since the plane passes through the origin, and since u and v are not parallel,

u and v span the plane and we have the vector equation s*u + t*v describing

the plane, where s and t are reals. Thus the parametric representation would be

x = s, y = -s + t and z = -t.