Let $f_1:\Bbb R\to\Bbb R^3$ and $f_2:\Bbb R\to \Bbb R^3$. The curves are given by: $f_1(t)=(t,t^2,t^3)$ and $f_2(t)=(e^t,\cos(t),\sqrt{1+t^2})$
So far, I represented each of the parametric variables as different entities (I named t=s for f2(t)), but now I am stuck regarding the system of equations. I am struggling to transform the system of equations into a matrix, because it contains an exponential, cosine and a squareroot.
I'd be glad if someone could help me out with this problem!(:
The given parametrizations produce the following equations:
$$t=e^s$$ $$t^2=e^{2s}=\cos s$$ $$t^3=e^{3s}=\sqrt{1+s^2}$$
For $s>0$ we have $2s>0$, and so $e^{2s}>1\ge\cos s$ i.e. $e^{2s}\ne\cos s$ - a contradiction.
For $s<0$ we have $3s<0$ and so $e^{3s}<1\le\sqrt{1+s^2}$ i.e. $e^{3s}\ne\sqrt{1+s^2}$ - a contradiction.
The only remaining value is $s=0$, which implies $t=1$, and gives you an intersection point $(1,1,1)$.