Finding if a path is tangent to a level surface

Question

The question is as followed:

Consider the path $r(t)=<t,2\sin2t,2\cos2t>$

TRUE/FALSE: the path $r(t)$ is tangent to the level surface $f(x,y,z)=x^2-y^2+z=1$

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I guess what's throwing me off is the lack of a specific point and the fact that it is a path equation with $t$ as a variable. I attempted this problem by parameterizing $r(t)$ with $x(t),y(t),z(t)$ and plugging that into $f(x,y,z)$ but that didn't lead me anywhere.

Answer 1

From my answer key:

FALSE. At $t= 0$, we have $r(0) =〈0, 2 \sin\theta, 2 \cos \theta〉=〈0,0,2〉$. However,$f<0,0,2> = 0^2−0^2+2 =2 \neq 1$. Thus the path is not contained in the level surface $f(x,y,z) = 1$.

Hope that helps someone.