Problem: Find the equation of a plane normal to the curve $(e^t, t, t^2)$ at the point $t=1$. Set the curve to $P(t)=(e^t, t, t^2)$, then taking the derivative twice we get $P''(t)=(e^t, 0, 2)$. Using $(X-P) \cdot N=0$ If $P''(1) = N$ and $P(1)=P$, then I get $ex+2z=e^2+2$. I'm pretty sure $P\cdot N=0$ should be met, but I'm not exactly sure where I went wrong. Is this the correct approach? I have realized with a few counterexamples that this derivation may not yield the correct normal. Any insights?
smokeypeat
asked Sep 28, 2017 at 4:25
smokeypeat smokeypeat
651 1 1 gold badge 7 7 silver badges 19 19 bronze badges
$\begingroup$ Why are you taking the derivative twice? $\endgroup$
Commented Sep 28, 2017 at 4:27
$\begingroup$ It is asking for the plane normal to the curve, not tangent. The thinking is the first derivative is tangent $\endgroup$
Commented Sep 28, 2017 at 4:34