Problem: If there is some subset defined on $\Lambda \times \mathbb{R}^3$ such that $\Lambda$ is a regular surface in $\mathbb{R}^3,$ then the subset $\Psi= \{(a,ka)\in \Lambda \times \mathbb{R}^3:k\in \mathbb{R}\}$ is a sub-manifold of $\Lambda \times \mathbb{R}^3.$
So this is the problem I am stuck with at the moment, I know that if one is a submanifold of another, then I have to show that the map $I$ is an injection where:
$I:\Psi \rightarrow \Lambda \times \mathbb{R}^3$
I am really stuck with this problem as I am not yet experienced with manifolds, can anyone show me how to do this so I can get the idea of how to do these problems?
EDIT: $(0,0,0)\notin \Lambda$
hint: the set $f^{-1}(0) \subseteq \Lambda\times\mathbb{R}^3$ for a smooth function $f:\Lambda\times\mathbb{R}^3 \rightarrow \mathbb{R}$ is a smooth manifold, take e.g. $f: (x,y) \mapsto kx-y$