Show that the parametric graphs intersects at the point $P=(2,1,3)$

Question

Show that the following graphs intersects at the point $P=(2,1,3)$: \begin{align} r_1(t)&=2e^{-t} \hat i+\cos t \hat j+(t^2+3) \hat k \\ r_2(t) &=(1-t) \hat i+t^2 \hat j+(t^3+4)\hat k \end{align} We have to do the following

$ 2e^{-t_1}=(1-t_2), \\ \cos t_1=t_2^2, \\ t_1^2+3=t_2^3+4$

If we show, $t_1=0 $ and $t_2=-1$, then we get

$r_1(0)=(2,1,3) \\ r_2(-1)=(2,1,3)$.

How to show that ?

Any help

Answer 1

That the problem gave you the putative point of intersection makes it very easy; all you had to do was to solve, say, $2e^{-t_1}=2=1-t_2$ to get a value for $t_1$ and $t_2$, then substitute in the other coordinates to see if they match. If the intersection had not been given it would have become much harder.