Question

I posted this one yesterday..anyone want to try a challenging question? Pls help!
http://openstudy.com/groups/mathematics#/groups/mathematics/updates/4e3d65a90b8bfc76a3f6c271

ind parametric equations for the tangent line to the curve of intersection of the paraboloid z=x^2 + y^2 and the ellipsoid 4x^2 + y^2 + z^2 = 9 at the point (-1,1,2).

Answer 1

x^2 +y^2 -z
grad_vector 1=<2x,2y,-1>
at that point <-2,2,-1>

4x^2+y^2+z^2=9
grad_1=<8x,2y,2z>
<-8,2,16>

<-2,2,-1> x <-8,2,16>
{34,40,12}
{17,20,6}

use that for line equation
x=-1+17t
y=1+20t
z=2+6t

Answer 2

The book gives
x=-1-10t
y=1-16t
z=2-12t

Answer 3

I think I found my mistake
4x^2+y^2+z^2=9
grad_1=<8x,2y,2z>
<-8,2,4> made mistake up there!
so cross product is between
<-8,2,4> and <-2,2,-1>
I get <-10,-16,-12>

so you are right,
x=-1-10t
y=1-16t
z=2-12t

Answer 4

Niice :))) You're the only one who even attempted this one in the last 2 days. Thanks you! I was trying to solve for the curve of intersection first and then...but didn't past the and then part lol.

Answer 5

I didn't get it when my prof explains it , I learned it after playing around on Mathematica

Answer 6

Here is better pic

Answer 7

Yeah...so tough to visualize. I'll learning this stuff independently from the MIT ocw site (and my old Calc text from my University days)

Answer 8

damn typos today lol. You wouldn't think english is my first (and only) language lol

Answer 9

hold on let me send you really intuitive plot that helped me

Answer 10

Where are you generating these graphs from?

Answer 11

Mathematica

Answer 12

Commercial software?

Answer 13

Yes, it is free for math majors, but unfortunately not for other , I have to pay

Answer 14

cool

Answer 15

One last question: how do you know what order to take in this cross product? Why <-8,2,4>x<-2,2,-1> and not <-2,2,-1>x<-8,2,4>?

Answer 16

It does not matter it will show you
<8,2,4>x<-2,2,-1>=<-10,-16,-12>
<-2,2,-1>x<-8,2,4>=<10,16,12>
this is same vector since they are scalar multiple(-1) of one another

Answer 17

I guess it just depends on which way we are counting the parameter "t". So t is negative on one side for <-10,-16,-12> but it is negative on the other side of t=0 for <10,16,12>. Makes sense

Answer 18

that works too

Answer 19

thanks again man...you've been a huge help!

Answer 20

I did MIT OCW,didn't remember that problem there

Answer 21

It's from my textbook problems set :)

Answer 22

i cant seem to focus on what the stated problem is asking for :)

Answer 23

So if you have you surface , we had to find to point of intersection of two plane

Answer 24

so basically they intersect at a curve

Answer 25

The paraboloid and the ellipsoid intersect in a surface which contains the point (-1,1,2). We want the tangent line to this surface at the point (-1,1,2). The tangent line will have three parametric equations (for x,y,z) which describe it. We want those parametric equations.

Answer 26

so the intersection of the two creates a conic on a plane; and the conic has a tangent at the given point .... now i see it :)

Answer 27

Well the surface of intersection is not on a plane. It isn't described by a linear relation. But yeah, we want the tangent line to this surface at that point.