1. ## Helix

Show that $\displaystyle \mathbf{R} = \cos{t} (\mathbf{i} - \mathbf{j}) + \sin{t} (\mathbf{i} + \mathbf{j}) + \frac{1}{2}t\mathbf{k}$ is a helix

I'm not sure what I have to show. I've already computed $\displaystyle \mathbf{R}'(t), \mathbf{R}''(t), \mathbf{T}(t)$ and $\displaystyle \kappa$

2. You're on the wrong track. You need to transform the equations you have into a set of equations that clearly define a helix.

Try rotating the coordinate axes clockwise by $\displaystyle 45^\circ$ around the z-axis.

The substitution is:

$\displaystyle \mathbf{i}=\frac{\sqrt{2}}{2}\mathbf{i}'+\frac{\sq rt{2}}{2}\mathbf{j}'$

$\displaystyle \mathbf{j}=-\frac{\sqrt{2}}{2}\mathbf{i}'+\frac{\sqrt{2}}{2}\m athbf{j}'$

and of course $\displaystyle \mathbf{k}=\mathbf{k}'$.

And you'll need to know that a helix is defined by a particle traveling at a constant rate in the z direction, and around a circle with constant angular speed in the x-y direction.

Post again if you're still having trouble.