For a curve defined parametrically by

the length of the curve, s, from point a to b is given by

if  are continuous on [a,b].

If you haven’t seen the formula before, it should be difficult to show. Just apply the Riemann sums with  via Pythagoras theorem.

Anyways, this formula turns out to be very useful in determining unit vectors functions, vectors which always have magnitude 1, which itself is needed to many concepts that follow.

Under close inspection, you’ll notice that the equation under the radical is the length of the tangent vectors and defined in the previous section.

where . And so we may now write the arc length as

We will now use this idea to investigate a typical curve to see what interesting results we get.

Suppose a curve C  is defined parametrically by

We first write the position vector of the curve. Remember, the position vector is a vector function that gives us a range of vectors each going from the origin to a point on the curve. We do this so we can use so-called ‘vector concepts’ like differentiation which would not have been possible if we stayed with the parametric equations above.

The position vector is

The tangent vector is

The length of the tangent vector is

And finally the length of the curve is

Here is when things get a little interesting. See, in vector differential calculus, we tend to develop concepts with the use of unit vectors. Now, in general, the tangent vector  will not have unit length for all t, that is a length of 1. But mathematicians have found out that if we change the parameter of the vector function from t to the arc length s, we obtain a vector function whose magnitude is always 1. This means, we aim to find a vector function in terms of the arc length of the curve.

As before, say we are given a position vector  of a curve defined parametrically by  for . Let  be the point  on the curve. We may think of this as the initial point of C. For , we define a function  by

which is the length of the part of C from  to .

                                                                                                
Since we have the function , we can rearrange this to give us . By substituting  into the parametric equations, we can now express the curve in terms of s,

L is the length of the curve. In terms of s, we now have position vector,

By using chain rule, we can different the position vector in terms of s, giving us

Remember again another formula from your high school calculus which is

we get

And finally

And there you have it, by expressing the tangent vector in terms of the arc length s, we will always get a unit vector. This is NOT to be confuse with the direction of the vector. Its direction is still variable depending, of course, on what the vector function is. Just remember that whatever vector, its length is always 1 or unit.