Step 1

Given Parabola

\(\displaystyle{y}={x}^{{{2}}}+{1}\)

The Vector valued function is

\(\overrightarrow{r}(t)=x(t)\hat{i}+y(t)\hat{j}\)

Step 2

In Parabola eqn, y is explicity defined in terms of x

\(\displaystyle\therefore{x}={t}\)

\(\displaystyle{y}={t}^{{{2}}}+{1}\)

Ans: \(\overrightarrow{r}(t)=t\hat{i}+(t^{2}+1)\hat{j}\)