Answer
$x=1+t,\ \ \ y=t$, $\ \ $ $z =t$
Work Step by Step
Differentiate $(\displaystyle \frac{d}{dt})$ each component function. to find the tangent vector at t.
$\mathrm{r}^{\prime}(t)=\langle e^{t}$, $\ \ te^{t}+e^{t}, \ \ 2t^{2}e^{t^{2}}+e^{t^{2}}\rangle$
Fot the point $(1, 0,0),$
find t from the parametric equations: $\quad\left\{\begin{array}{lll}
1=e^{t} & \Rightarrow t=0 & \\
0=te^{t} & \Rightarrow 0=0 & (ok)\\
0=te^{t^{2}} & \Rightarrow 0=0 & (ok)
\end{array}\right.$
so $\mathrm{r}(0)=(1, 0,0).$
A tangent vector at $(1, 0,0)$ is
$\mathrm{r}^{\prime}(0)=\langle e^{0},0+e^{0},0+e^{0}\rangle=\langle 1,1,1\rangle$
The tangent line contains $(1, 0,0)$ and is parallel to the vector $\langle 1,1,1\rangle$.
Its parametric equations are (see sec 12-5):
$x=1+t,\ \ \ y=t$, $\ \ $ $z =t$
