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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 vectorsi + j + - kandi - 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 + - kand b =i - j + kwhich 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 + kand 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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# Parametric Surfaces and Their Areas

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