The vector function is, r(t) = ![\bold{ < t,2t^2,9t^2+4t^4 > }](https://tex.z-dn.net/?f=%5Cbold%7B%20%3C%20t%2C2t%5E2%2C9t%5E2%2B4t%5E4%20%3E%20%7D)
Given two surfaces for which the vector function corresponding to the intersection of the two need to be found.
First surface is the paraboloid, ![z=9x^2+y^2](https://tex.z-dn.net/?f=z%3D9x%5E2%2By%5E2)
Second equation is of the parabolic cylinder, ![y=2x^2](https://tex.z-dn.net/?f=y%3D2x%5E2)
Now to find the intersection of these surfaces, we change these equations into its parametrical representations.
Let x = t
Then, from the equation of parabolic cylinder,
.
Now substituting x and y into the equation of the paraboloid, we get,
![z=9t^2+(2t^2)^2 = 9t^2+4t^4](https://tex.z-dn.net/?f=z%3D9t%5E2%2B%282t%5E2%29%5E2%20%3D%209t%5E2%2B4t%5E4)
Now the vector function, r(t) = <x, y, z>
So r(t) = ![\bold{ < t,2t^2,9t^2+4t^4 > }](https://tex.z-dn.net/?f=%5Cbold%7B%20%3C%20t%2C2t%5E2%2C9t%5E2%2B4t%5E4%20%3E%20%7D)
Learn more about vector functions at brainly.com/question/28479805
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