Normal Plane to a Curve.

1. Oct 27, 2009

Septcanmat

1. The problem statement, all variables and given/known data

At what point on the curve ⃗r(t) = (t^3, 3t, t^4) is the normal plane parallel to the plane 3x + 3y − 4z = 9 (the normal plane is the plane through the point ⃗r(t) which is normal to ⃗r′(t))

2. Relevant equations

I'm not really sure.

3. The attempt at a solution
(6t)(x-t^3) + (0)(y-3t) + (8t)(z-t^4) = 0

But that got me nowhere.

2. Oct 28, 2009

lanedance

i'm not sure what you attempted there...

first find vector normal to the plane given, then find the tangent vector of the curve.. and have a think about how they will be related

3. Oct 28, 2009

Septcanmat

I read somewhere that that would be the equation of a normal plane to a curve. But it didn't work.

I did what you said, and they'll be related in that they'll be parallel vectors. I tried doing what you said and setting them equal to each other, but I just got equations for t that seem insolvable. ( for instance, 0= 8t^4 + 9 +16t^6)

4. Oct 28, 2009

lanedance

that's not what i get, it works out ok... i'm not sure how you get the higher powers of t in your equation either

what do you get for the normal to the plane & for the tangent vector?

5. Oct 28, 2009

Septcanmat

The normal is the gradient, so I got (3,3,-4). And I got (3t^2,3,4t^3)/sqrt(9t^4+9+16t^6) for the tangent vector.

6. Oct 28, 2009

lanedance

both look good, but I see you are normalising the tangent vector to length 1, that's not needed here, as you just need to know when its parallel

so if p is normal to the plane, t is the tangent, you just need to know when t = c.p for any constant c, which shows they are parallel. This should lead to a reasonably easy equation set if you don't normalise the vector

Last edited: Oct 28, 2009
7. Oct 28, 2009

Septcanmat

Well alright then, lol. Thanks guys, got it all figured out now. The answer is (-1,-3,1). Or rather, I'm assuming that that's the correct answer because it's what I got and it matches up with one of the multiple choice options :P

8. Oct 28, 2009

lanedance

yep thats what i get