- #1
wubie
[SOLVED] Parametric Surfaces and Their Areas
Hello,
I am having problems visualizing a concept. First I will post my question as it is given in Jame's Stewart's Fourth Edition Multivariable Calculus text, Chapter 17, section 6, question 17.
Find a parametric representation for the given surface.
(a) The plane that passes through the point (1,2,-3) and contains the two vectors i + j + - k and i - j + k .
Now I know that vector representation in the solution can be written as
r(u,v) = rsub0 + ua + vb
where a = i + j + - k and b = i - j + k which becomes
r(u,v) = <1,2,-3> + u<1,1,-1> + v<1,-1,1>
which would produce parametric equations
x = 1 + u + v,
y = 2 + u -v,
z = -3 -u + v.
But what I am wondering what if I let a = i - j + k and b = i + j + - k . Then I would have
r(u,v) = <1,2,-3> + u<1,-1,1> + v<1,1,-1>
which would produce different parametric equations than the first.
x = 1 + u + v,
y = 2 - u + v,
z = -3 + u - v.
Now intuitively I think this is just as valid as the first. Is it though?
Any help / input is appreciated. Thankyou.
I'm back and the more I think about it and fool around with it I believe it is not possible to have two vector equations representing a plane with the above criteria. If that is the case then how do I determine which vector is multiplied by the parameter u and which vector is multiplied by the parameter v? This is the part that is confusing me.
Hello,
I am having problems visualizing a concept. First I will post my question as it is given in Jame's Stewart's Fourth Edition Multivariable Calculus text, Chapter 17, section 6, question 17.
Find a parametric representation for the given surface.
(a) The plane that passes through the point (1,2,-3) and contains the two vectors i + j + - k and i - j + k .
Now I know that vector representation in the solution can be written as
r(u,v) = rsub0 + ua + vb
where a = i + j + - k and b = i - j + k which becomes
r(u,v) = <1,2,-3> + u<1,1,-1> + v<1,-1,1>
which would produce parametric equations
x = 1 + u + v,
y = 2 + u -v,
z = -3 -u + v.
But what I am wondering what if I let a = i - j + k and b = i + j + - k . Then I would have
r(u,v) = <1,2,-3> + u<1,-1,1> + v<1,1,-1>
which would produce different parametric equations than the first.
x = 1 + u + v,
y = 2 - u + v,
z = -3 + u - v.
Now intuitively I think this is just as valid as the first. Is it though?
Any help / input is appreciated. Thankyou.
I'm back and the more I think about it and fool around with it I believe it is not possible to have two vector equations representing a plane with the above criteria. If that is the case then how do I determine which vector is multiplied by the parameter u and which vector is multiplied by the parameter v? This is the part that is confusing me.
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