Find the equation of the plane normal to $\displaystyle r(t) = <e^t sin(pi/2 * t), e^t cos(pi/2 *t), t^2>$ when $\displaystyle t = 1$.
Sorry not sure how to display pi.
Where do I start?
A vector tangent to r(t) is dr/dt. Evaluate at t = 1 and you have the normal vector to the plane.
r(1) gives the position vector of a point in the plane.
So you have a normal to the plane and a point in the plane.
The equation of a plane is ax + by + cz = d where <a, b, c> is a normal to the plane and d is found by substituting a known point into ax + by + cz = d (once you've substituted the values of a, b and c of course).
I'm assuming that $\displaystyle \textbf{r}(t)=\left<e^t\sin(\pi t/2),e^t\cos(\pi t/2),t^2\right>$ is meant to represent the x,y and z co-ordinates (respectively).
If that is the case then, when t=1, we have $\displaystyle \textbf{r}(t)=\left<e,0,1\right>$.
Recall the dot product - $\displaystyle \textbf{a}.\textbf{b}=|\textbf{a}||\textbf{b}|\cos (\theta)$, where $\displaystyle \theta$ is the angle between the two vectors.
Note that, if two vectors are orthogonal then $\displaystyle \theta=\pi/2$ and so $\displaystyle \textbf{a}.\textbf{b}=|\textbf{a}||\textbf{b}|\cos (\pi/2)=0$ (as the cosine of $\displaystyle \pi/2$ is zero).
Hence, you want to find another vector, call this $\displaystyle \textbf{b}=\left<b_1,b_2,b_3\right>$ and find $\displaystyle \textbf{r}(1).\textbf{b}=0$.
In other, words, you are looking for solutions to the equation $\displaystyle eb_1+0*b_2+b_3=eb_1+b_3=0$. This implies that $\displaystyle b_1/b_3=-e$ and hence there are infinitely many solutions of this form
Also, as a sidenote, if t is not constant, you get a really interesting kind of geometry based on spirals. It 'looks' a little bit like making your way from the top of a trumpet to the bottom (or vice versa, depending) only going around the inside of the horn (I think that's what it's called)
Hope that helps
Keith
So if I understand this right, I take the the derivative of the vector and evaluate it at t=1 then evaluate just the vector its self at t=1. <a,b,c> will be the derivative of the vector evaluated at t=1 and the point I get from the vector I substutute for (x,y,z). Is this correct?