Is there a formula to find the intersection of a plane and a curve at a given point?

1. Aug 27, 2007

lldaniel144

1. The problem statement, all variables and given/known data
Find the intersection of the x_1x_2 plane and the normal plane to the curve
x= (cos(t)e_1 + (sin(t))e_2 + (t)e_3

At the point t = pi/2

2. Relevant equations

I have looked everywhere for a formula or an example for this and cannot find one? Can anyone help me as to what I should be looking up, if there is a formula, or a hint on the method I should try.

Thanks

2. Aug 27, 2007

HallsofIvy

Staff Emeritus
First, determine the planes! The tangent vector to cos(t)e_1 + (sin(t))e_2 + (t)e_3
is -sin(t)e_1+ cos(t)e_2+ e_3 and at pi/2 that is -e_1+ e_3. Of course, at pi/2, the curve goes through the point e_2+(pi/2)e_3.

The equation of a plane with normal vector -e_1+ e_3 containing point (0,1,pi/2) is, of course, -x_1+ x_3- pi/2= 0 or x_3- x_1= pi/2.

I assume you know that the equation of the x_1x_2plane is x_3= 0.

Find all points that satisfy x_3- x_1= pi/2 and x_3= 0. The intersection of two planes is, of course, a line.

3. Aug 27, 2007

matt grime

Why did you look for the formula? Such things are easy to derive for yourself.