Helicoid in $\mathbb{R}^4$

Question

I'm considering a parametric equation of helicoid in $\mathbb{R}^4$ is $X(u,v)=(u \cos v, u \sin v,v,v).$ Now I can calculate two unit tangent vectors $t_1=\dfrac{X_u}{\vert X_u\vert}=(\cos u, \sin u, 0,0) $ and $t_2=\dfrac{X_v}{\vert X_v \vert}=\dfrac{1}{\sqrt{u^2+2}}(-u \sin v, u \cos v, 1, 1)$. The problem is how to find 2 unit normal vectors $n_1, n_2$? I tried and find that $n_1= \dfrac{(0,0,1.-1)}{\sqrt{2}}$ is ok, but the second unit normal vector is more difficult.

Answer 1

Of course, this is just "the usual" helicoid sitting in the hyperplane $x_3=x_4$. The normal vector you found is the normal vector to that hyperplane. If you review the computation for "the usual" helicoid in $\Bbb R^3$, you have the (non-unit length) normal $(\sin v,-\cos v, u)$. So you might guess the vector $(\sin v,-\cos v, u/2, u/2)$. Note that this is orthogonal to both $t_j$ and to $n_1$.