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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.
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