Which equation is the proper equation to find the tangent line to a curve in space?

Question

From my understanding, to write an equation of a line in 3D all you need is point and direction vector. So for a tangent line I assumed all you need is the point of tangency and the tangent direction vector at that point. So the question asks the tangent line at $t=$something. So my equation would look like $r(t) + t\cdot r'(t)$.
So use the terminal point of the vector $r(t_0)$ and shoot a line through the direction given by the vector $r'(t)$. But the equation in my book says that the tangent line equation is $r(t_0)+(t-t_0)\cdot r'(t)$. I do not understand where the $t-t_0$ comes from and why my equation would be wrong