tangent planes: r(t)=<3t,1-t^2,t^2>, r(u)=<1+u^2,2u^3,u+2>

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lislr8

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I'm having some trouble finding a tangent plane that passes through 2 vector curves at the point (2,1,3). If I'm given for example r(t)=<3t,1-t^2,t^2> and r(u)=<1+u^2,2u^3, u+1>? I have no idea where to start.

Thanks
lislr
 
lislr8 said:
I'm having some trouble finding a tangent plane that passes through 2 vector curves at the point (2,1,3). If I'm given for example r(t)=<3t,1-t^2,t^2> and r(u)=<1+u^2,2u^3, u+1>? I have no idea where to start.

Thanks
lislr
Click to expand...
Do you know how to find the normal vector for the tangent plane?

One necessary condition for common tangent plane would be that the directional cosines of the tangent planes of the curves would be same.
 
lislr8 said:
I'm having some trouble finding a tangent plane that passes through 2 vector curves at the point (2,1,3). If I'm given for example r(t)=<3t,1-t^2,t^2> and r(u)=<1+u^2,2u^3, u+1>? I have no idea where to start.
Click to expand...
You got a real problem in that the point (2,1,3) is on neither of those two curves.
Did you copy the question correctly?
 
sorry I was trying to alter the question a little but apparently that didn't work out. It's actually r(t)=2+3t, 1-t^2, 3-4t+t^2>
u(t)= <1+u^2, 2u^3-1, 2u+1> at the same point above.
 
Back to my original "prop"....

Do you know how to find the normal vector for the tangent plane?

One necessary condition for common tangent plane would be that the directional cosines of the tangent planes of the curves would be same.
 
I don't think I know how to find a normal vector to tangent plane. If there was an equation of a plane I could find a normal vector to it. So like x+y+z=1 the normal is just <1,1,1,>. How do I do that without a plane to begin with?

What do you mean by directional cosines?
 
Here is the basic difficulty with the statement.
Curves do not have tangent planes, they have tangent lines.
As restated the point belongs to both curves: \(\displaystyle r_1 (0) = (2,1,3) = r_2 (1)\).
Now there is a plane that contains the tangent lines to both curves at the given point.
Its normal is \(\displaystyle \;\left[ {r_1 } \right]^\prime (0) \times \left[ {r_2 } \right]^\prime (1)\).
Now you can finish.
 
Thanks for the help everyone. I think I'm going to do this the long way and find a tangent line to each curve. Then I'll find the cross product of the two tangent vectors and I'll write a equation for a line with the normal vector and point given.
 
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