The discussion revolves around determining whether specific points, P(7,10,4) and Q(5,22,5), lie on a parametric representation of a surface defined by the equation r(u,v)=<2u+3v, 1+5u-v, 2+u+v>. Participants are exploring the implications of the parametric equations and their relationship to the points in question.
The discussion has seen various interpretations of the problem, with some participants expressing confusion over the terminology used in the original statement. While one participant claims to have solved the problem, others are still exploring the implications of the parametric equations and how they relate to the points in question.
There is a noted discrepancy regarding the classification of the equation as a line versus a surface, leading to questions about the proper approach to determining if the points lie on the described geometric entity. Some participants mention the need for clarity on the definitions and context provided in the original problem statement.
KnowledgeisPo said:Homework Statement
Determine whether the points P(7,10,4) and Q(5,22,5) lie on the given line:
Homework Equations
r(u,v)=<2u+3v, 1+5u-v, 2+u+v>The Attempt at a Solution
x=x(u,v)=2u+3v, y=1+5u-v, z=2+u+v
x+y+z=8u+3v+3
KnowledgeisPo said:I see what you are saying. Keep in mind this is Parametric Surfaces and Their Areas. That was all the given information, and that is how the wording is transcribed in the book. Perhaps P and Q corresponds to the vector itself, and i and j of the vector are to correspond to P and Q respectively, for z to confirm proper coordinates. The exact wording is as follows:
"Determine whether the points P and Q lie on the given line.
1. r(u,v) = <2u+3v, 1+5u-v, 2+u+v>
P(7,10,4), Q(5,22,5)"
KnowledgeisPo said:Thank you for your time, and sorry for any inconvenience, as I solved it myself. I set the partial vectors equal to the respective points, and used the simple Z partial to put v in terms of u. Upon doing that, I substituted that in for the X partial to get u and v coordinates at the point. Upon applying the u and v coordinates to the Y partial, the equation of the Y partial equaled to the points came out true and false for each of the points. The true and false calculations matched the answer in the back of the book.
Thank you, and if you request, I will post the solution.
Yes, this is a little bit why I didn't understand the book.LCKurtz said:But that R(u,v) is a surface, not a line. It would make sense to check whether those two points are on the surface.
And whether or not the points lie on the surface has nothing to do with the partial derivatives of R. So at this point, I still have no idea of what you are trying to do.
As has been said, that is a surface (actually a plane), not a line. To determine if the point (7, 10, 4) lies on that plane, just set 2u+ 3v= 7, 1+ 5u- v= 10 and solve for u and v. If those values of u and v also satisfy 2+ u+ v= 4, then the point is on the plane."Determine whether the points P and Q lie on the given line.
1. r(u,v) = <2u+3v, 1+5u-v, 2+u+v>
P(7,10,4), Q(5,22,5)"
KnowledgeisPo said:Yes. As I said a few posts, I have solved it. I was aware that multivariable parametric surfaces are surfaces. However, was I supposed to not say what the book said? Thank you again for the assistance.
KnowledgeisPo said:...and if you request, I will post the solution.