Tangent, Normal and Binormal Vectors

In this part we want to look at an application of derivatives for vector functions.  In fact, there are a couple of applications, but they all come back to requiring the first one.

In the past we have employed the fact that the derivative of a function was the slope of the tangent line. Along with vector functions we obtain exactly similar result, along with single exception.

There is a vector function, r^→ (t) , we call ^→r′ (t) the tangent vector specified by it exists and provided ^→r′ (t) ≠ 0 . After that the tangent line to ^→r (t) at P is the line that passes via the point P and is parallel to the tangent vector, ^→r′ (t). 

Notice: we really do need to require r?′ (t) ≠ 0 to have a tangent vector.  Whether we had ^→r′(t) = 0^→ we would have a vector that had no magnitude and thus could not give us the direction of the tangent.