I have the following problem in $E_3^*$
I have to find a parametric equation of a line passing through the point $M(3,1,2,1)$ and the infinite point of of a line $l$ formed from two planes $\alpha$ and $\beta$.
$$ \alpha: x+y=0\\\beta: y+z-t=0 $$
What I've got:
$$ \left\{ \begin{array}{c} x+y=0 \\ y+z-t=0 \\ t=0 \end{array} \right. $$
From there I find the coordinates of the point $U_l$ to be $(-1,1,-1,0)$ and the parametric equation of a line passing $U_l$ and M to be
$$ \left\{ \begin{array}{c} x=3\lambda-\mu \\ y=1\lambda+\mu \\ z=2\lambda-1\mu\\ t=1\lambda+0\mu \end{array} \right. $$
Is this the correct approach?
Thanks in advance!