stereographic projection, again

The problem:

Let $\displaystyle \phi:\mathbb{R}^3 \rightarrow S^3$ be the inverse of stereographic projection from $\displaystyle S^3-\{(0, 0, 0, 1)\}$ to the equatorial hiperplane $\displaystyle x_4=0$. Show that for every vector $\displaystyle p \in \mathbb{R}^3$ exists $\displaystyle \lambda(p) \in \mathbb{R}^3$ such that

$\displaystyle ||d\phi(\textbf{v})||= \lambda (p) ||\textbf{v}||, \forall \textbf{v} \in {\mathbb{R}^3}_p$.

The solution must rely on the following hint: the vector part of $\displaystyle d \phi (\textbf{v})$ is $\displaystyle (d/dt)|_{t=0} \phi (p + tv)$, where $\displaystyle \textbf{v}=(p, v)$.

I can't even get this vector part right..

As always, thanks for all the hints and help.

I will continue to work on it and post if I get anything.

(If it matters, it's not homework or exam question, just exercise.)