I have to determine for which values of $x$ that the velocity vector is orthogonal to the acceleration vector, the position is given by:
$(3 \cos(t), - \sin(3t), 2t^3 - t^2)$,
I then use that $u \cdot v = 0$ if $u$ and $v$ is orthogonal. The dot product of the acceleration and velocity gives the following equation (if I have done the differentiation correctly):
$9\sin\left(t\right)\cos\left(t\right)-27\cos\left(3t\right)\sin\left(3t\right)+4t\left(18t^2-9t+1\right) = 0$.
$t = 0$ (or position $x = (3,0,0)$) is clearly a root but i can't find the other two. Is there a general way of computing roots of this sort of function analytically?