- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
14.7 Exercises
- Consider the helix described by the vector equation $\textbf{r}(t) = a\cos \omega t\mathbf{i} + a\sin \omega t\mathbf{j} + b\omega t\,\mathbf{k},$ where $\omega$ is a positive constant. Prove that the tangent line makes a constant angle with the $z$-axis and that the cosine of this angle is $b/\sqrt{a^2 + b^2}.$
- First, we find the equation of the tangent vector $\mathbf{v}(t) = \mathbf{r}'(t):$ $$\mathbf{v}(t) = -a\omega\,\sin \omega t\,\mathbf{i} + a\omega \cos \omega t\,\mathbf{j} + b \omega\,\mathbf{k}$$ Then, we take its dot product with $\mathbf{k},$ which is equal to $b\,\omega.$ Written another way, $$b\,\omega = \|\mathbf{v}(t)\|\,\|\mathbf{k}\|\cos \theta$$ Where $\theta$ is the angle between $\mathbf{v}$ and $\mathbf{k}.$ Evaluating the norm of $\mathbf{v}(t)$ gives us $\omega\,\left(a^2 + b^2\right)^{1/2},$ and of course the norm of the unit vector $\mathbf{k}$ is 1. Combining these terms, we show that the angle $\theta$ is constant, with $$\cos \theta = \frac{b}{\sqrt{a^2 + b^2}}\quad \blacksquare$$