MAIN CONCEPTS
, representing the acceleration vector.
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Since T is the unit tangent vector and N is the unit normal vector, we can now write the acceleration vector as such.

We shall now attempt to proof the above. You need to have a good grasp of differentiating products of real-value functions and vector functions.

First, notice that in terms of a parameter t, we have

which follows to

Then

since .

But from the previous section, we defined the unit normal vector as , and so , giving us

Looking at this equation carefully, notice that the tangential and normal vectors are both of unit length, meaning to say that the magnitude of these vectors is dictated by the scalar functions before them.

Thinking along these lines, the scalar  is called the tangential component of the acceleration and is denoted by . The scalar  is called the normal component of the acceleration and is denoted by . With these definitions, another way to write the acceleration vector is

I would like to kindly remind the reader that this is another way of writing the acceleration vector instead of take the first derivative of the velocity vector. As you shall see in the later section, which way the reader finds the acceleration vector should be based on the convenience. Usually when the situation starts out with the position vector in terms of t, then taking the derivatives is desirable. If the position vector is in terms of the arc length, then this method should be more suitable.