In this section we want to look at an application of derivatives for vector functions. Actually, there are a couple of applications, but they all come back to needing the first one.
With vector functions we get exactly the same result, with one exception. While, the components of the unit tangent vector can be somewhat messy on occasion there are times when we will need to use the unit tangent vector instead of the tangent vector.
First, we could have used the unit tangent vector had we wanted to for the parallel vector.
However, that would have made for a more complicated equation for the tangent line. Do not get excited about that.
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Next, we need to talk about the unit normal and the binormal vectors. The unit normal is orthogonal or normal, or perpendicular to the unit tangent vector and hence to the curve as well.
They will show up with some regularity in several Calculus III topics. The definition of the unit normal vector always seems a little mysterious when you first see it.
It follows directly from the following fact. To prove this fact is pretty simple. From the fact statement and the relationship between the magnitude of a vector and the dot product we have the following.
Also, recalling the fact from the previous section about differentiating a dot product we see that,. The definition of the unit normal then falls directly from this. Because the binormal vector is defined to be the cross product of the unit tangent and unit normal vector we then know that the binormal vector is orthogonal to both the tangent vector and the normal vector.
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Here is the tangent vector to the curve. Show Solution We first need the unit tangent vector so first get the tangent vector and its magnitude.