[Math] How to parametrize the curve of intersection of two surfaces in $\Bbb R^3$

Question

I have to parametrize the curve of intersection of two surfaces. The surfaces are:

$$y^2 = z \text{ and } x + y = 4$$

Could someone please show me how to do this step by step?
Thanks.

Answer 1

Let $y=t$. Then $x=4-t$ and $z=t^2$ and so $\vec r(t)= (x(t), y(t), z(t))= (4-t, t, t^2)$, $t\in\mathbf R$.