Apologies if the title is sort of confusing. This is a HW problem, and I am stuck at exactly the magnitude of a vector.
\begin{array}{l}{\text { Problem 1. Let } \beta \text { be a unit speed cylindrical helix with } \beta^{\prime}(t) \cdot u=\cos \theta \text { for fixed } \theta} \\ {\text { and } u . \text { Consider the cross sectional curve }}\end{array}
$$ \gamma(t)=\beta(t)-t \cos (\theta) u $$
$\text { (1) Prove that } \gamma \text { lies in a plane orthogonal to } u$
I claim that $\gamma'(t) \cdot u = 0$ (to prove that it is orthogonal), and that claim simplifies to $\cos \theta - \cos \theta \cdot |u|^2 = 0 $. I obviously want $u$ to be a unit vector. $\beta^{\prime}(t) \cdot u=\cos \theta$ looks suspiciously like a property inherent to helixes, so how can I show $u$ to be of unit magnitude?
So in the end this seems like I did not read the problem properly. Yes, I can assume $u$ to be a unit vector by the phrasing and the definition of cylindrical helixes: see cmk's comments for more details.