How to represent this parametrically? For the purpose of solving a problem involving manifolds, I want to know how to represent this situation...
I have an  $S^2$ sphere $(x^2 +y^2 + z^2 =1)$ and a point $(a,b)$ in the plane $\mathbb{R}^2$. I want to connect it with the north pole of the sphere $(0,0,1)$ by a line to compute its intersection with $S^2$.
How do I find the equation of this line? 

The line goes through the points:
$$P_1(0,0,1)$$
$$P_2(a,b,0)$$
The direction vector, from $P_1$ to $P_2$ is, $\langle a-0,b-0,0-1 \rangle$. This is $\langle a,b,-1 \rangle$. Then the equation of the line is given by the position vector function,
$$\vec r(t)=\langle a,b,-1 \rangle t+\langle 0,0,1 \rangle$$
So we may parametrize as follows,
$$x=at$$
$$y=bt$$
$$z=1-t$$
