Find the tangent line

Question

Find the tangent line to the curve $\displaystyle r(t)= \left(3\ln t,2t-3,\frac1t \right)$ at $t=1.$

I know that I have to find the derivative of $r(t)$ but I do not know the following steps.

Answer 1

The tangent line to a parametrized curve at $t_0$ is given by $r(t_0)+\lambda r'(t_0), \lambda\in\Bbb{R}.$ In your case, it is the line $(0,-1,1)+\lambda(3,2,-1)=(3\lambda,-1+2\lambda,1-\lambda),\lambda\in \Bbb{R}.$