Parameterized tangent line to a parameterized curve

[Cauchy1789]

Homework Statement

I seem to remember that a parameterized a(t) curve in \mathbb{R}^3 that one can construct the tangent from the slope of a'(t) and the curve itself.

such that the tangent line L = a(t) + s * a'(t) to a. This is supposedly a straight line in \mathbb{R}^3.
To make a long question. What theorem allows me to construct the tangent in such a way? confused:

 

[tiny-tim]

I seem to remember that a parameterized a(t) curve in \mathbb{R}^3 that one can construct the tangent from the slope of a'(t) and the curve itself.

such that the tangent line L = a(t) + s * a'(t) to a. This is supposedly a straight line in \mathbb{R}^3.
To make a long question. What theorem allows me to construct the tangent in such a way? confused:

Hi Cauchy1789!

It's the theorem that says that the slope of the tangent euqals the derivative …

so, for fixed t, L(s) = s * a'(t) + constant …

and the constant has to be a(t) because L(0) = a(t).