Keeping in line with the previous problem, we will now see some methods in calculating the unit tangent and unit normal explicitly.
VECTOR DIFFERENTIAL CALCULUS >
FINDING UNIT TANGENT AND UNIT NORMAL
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MULTIPLE APPROACHES TO PROBLEMS.1
UNIT BINORMAL AND TORSION
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CONCEPT
We will continue from the previous lesson by looking at the same problem. To recall, we are given the position vector
And found the velocity, acceleration vectors and curvature to be
We will now explicitly find the unit tangent and unit normal vectors using clever manipulation of differentiating rules. By chain rule, we can express the unit tangent as follows,
To calculate the unit normal, we perform
Take note that We’ll briefly introduce another vector called the binormal vector. The binormal vector is defined by
All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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and 
and 
are orthogonal unit vectors.
is a unit vector orthogonal to the plane of 
and 
. The vectors 
form a right-handed coordinate system at each point on the curve and these three vectors is called the frenet frame which will be talked about in greater detail in the subsequent lesson.
,
and 
exist and thus there’s a frenet frame.
