We will now continue our definitions by talking about the last mutually perpendicular vector in the frenet frame, that is, the unit binormal.
In doing so, we will introduce the function torsion given by
, a important function in describing curves as we'll see in later lessons. VECTOR DIFFERENTIAL CALCULUS >
UNIT BINORMAL AND TORSION.2
May want to check out:
UNIT BINORMAL AND TORSION.1
FINDING UNIT TANGENT AND UNIT NORMAL.1
INTERPRETATION OF TORSION
VIDEO LECTURE part 2
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CONCEPT
At any point on the curve
 where  and  are defined, a third unit vector, the unit binormal  is defined by the formula
With the three unit vectors, To find out the directions of the vectors, take your hand and form a coordinate axis with your index, middle and thumb, all three being mutually perpendicular. Point your index finder in the direction of the first vector, your middle in that of the second, and your thumb will determine the direction of the third vector. Which direction the last vector will ultimately point depends on whether you are using your left or right hand. I should think it is obvious that when we specify the vectors in parenthesis, the order is important. The basis of
Bearing in mind the directions, note that
We now wish to formulate some results from this frenet frame and the process we go about doing it is by differentiation. Just like how differentiating the position vector gives us the velocity vector, by differentiating these vectors, taking into account all such rules, we get a new set of equations. From
we differentiate with respect to s,
And we conclude from here that However, there’s also another equation from which we can differentiate vector
which tells us that
Torsion given by the function
Torsion measures the degree of twisting that the curve exhibits near a point, that is to say, the extent to which the curve fails to be planar. It may be positive or negative depending on which direction the curve is twisting. A geometrical interpretation follows in the next lesson. All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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and is parallel to 
.
, they constitute a right-handed basis of mutually perpendicular unit vectors like the standard basis 
, the only difference being that it is right-handed.
is called the frenet frame.
is perpendicular to 
. A result we will use later.
, namely our definition 
. Doing so gives us
is also perpendicular to 
. Now, here comes the tricky part. At first we seem to have new derivatives 
and 
which we have no idea about. But currently their direction is more important than their meaning. Imagine that vector 
is fixed in one direction. Our previous equation tells us that we can ‘swivel’ 
and 
round vector 
as long as they are mutually perpendicular. At this juncture, we make use of our previous equation that tells us 
is perpendicular to 
, and as we know 
and 
are components from the frenet frame, the only possibility is that 
is parallel to 
which leads nicely to our definition of torsion. 
