How can I check my answer?

[Gavriell]

How can I check my answer? can I just plug t into the final parametric equation and check if I get the same given points?

 

[Steven G]

The 1st equation in which you solved for t also has the result of t=-2. Will this change anything?

 

[Gavriell]

The 1st equation in which you solved for t also has the result of t=-2. Will this change anything?

Yes, -2 does not satisfy the other two equations.

 

[pka]

The 1st equation in which you solved for t also has the result of t=-2. Will this change anything?

$\displaystyle -2$ is not in the domain of $\displaystyle \vec{r}$, because $\displaystyle \log(t+1)-\exp(t)$ would not exist for $\displaystyle t=-2$.

 

[Gavriell]

$\displaystyle -2$ is not in the domain of $\displaystyle \vec{r}$, because $\displaystyle \log(t+1)-\exp(t)$ would not exist for $\displaystyle t=-2$.

does that mean that the answer is correct?

 

[pka]

does that mean that the answer is correct?

Yes! As far as I can follow your workings.
It should be $\displaystyle \vec{v}(0)+t~\vec{v}~'(0)$