#### Answer

$x=1+t ; \\ y =t ; \\ z=-t$

#### Work Step by Step

$v(t)=\dfrac{dr}{dt}= e^t i+\cos t j -\dfrac{k}{1-t}$
Set $t=0$
Then, we have: $v(t)=\dfrac{dr}{dt}= e^{0} i+\cos (0) j -\dfrac{k}{1-0}=i+j-k$
and $r(0)=e^{0} i+\sin (0) j +\ln (1-0) k =i$
Thus, the parametric equations of the line at $(1,0,0)$ are:
$x=1+t ; \\ y =t ; \\ z=-t$