Tangent, normal, and binormal vectors

Tangent, normal, and binormal vectors

Lessons

Notes:

Tangent & Unit Tangent Vectors
For a vector function r(t)r(t), we call r(t)r'(t) the tangent vector where r(t)0r'(t) \neq 0. In a sense, r(t)r'(t) is a tangent line to r(t)r(t) at point PP.
tangent vector
The unit tangent vector is calculated by:

T(t)=r(t)r(t)T(t) = \frac{r'(t)}{||r'(t)||}

Where r(t)0 r'(t) \neq 0 . The unit tangent vector specifically has a length of one.
unit tangent vector
Unit Normal Vector
A unit normal vector N(t)N(t) is a vector that is orthogonal (or perpendicular) to the unit tangent vector T(t)T(t) and to the vector r(t)r(t).
unit normal vector
It is calculated by:

N(t)=T(t)T(t)N(t) = \frac{T'(t)}{||T'(t)||}

Binormal Vector
A binormal vector B(t)B(t) is a vector that is both orthogonal to the unit normal vector N(t)N(t) and unit tangent vector T(t)T(t). We can calculate B(t)B(t) by taking the cross product of N(t)N(t) and T(t)T(t).

B(t)=T(t)×N(t)B(t) = T(t) \times N(t)

  • Introduction
    Tangent, Normal, & Binormal Vectors Overview:
    a)
    Tangent & Unit Tangent vectors
    • r(t)r(t) \to vector function
    • r(t)r'(t) \to tangent to the vector function
    • T(t)T(t) \to unit vector tangent to the vector function
    • T(t)=r(t)r(t)T(t) = \frac{r'(t)}{||r'(t)||}

    b)
    Unit Normal Vector
    • N(t)N(t) \toorthogonal to r(t)r'(t) and T(t)T(t)
    • N(t)=T(t)T(t)N(t) = \frac{T'(t)}{||T'(t)||}
    • A unit vector (length of 1)

    c)
    Binormal Vector
    • B(t)B(t) \to orthogonal to both T(t)T(t) and N(t)N(t)
    • Calculated by taking the cross product T(t)×N(t)T(t) \times N(t)


  • 1.
    Finding the Tangent Vector and Line
    Find the tangent line to the vector function r(t)=<t32,2+2t2,t>r(t)= \lt t^3-2, 2+2t^2, t\gt at t=1t=1.

  • 2.
    Find the tangent line to the vector function r(t)=<cost,sint,tant>r(t)= \lt\cos t, \sin t, \tan t\gt at t=πt= \pi.

  • 3.
    Finding the Unit Tangent Vector Function
    Find the unit tangent vector for the given vector function:

    r(t)=<e4t,t2+3,et> r(t) = \lt e^{4t}, t^2 + 3, e^t \gt


  • 4.
    Finding the Unit Normal Vector
    Find the unit normal vector for the given vector function:

    r(t)=<sint,cost,2t> r(t) = \lt \sin t, \cos t, 2t \gt


  • 5.
    Find the unit normal vector for the given vector function:

    r(t)=<etsint,etcost,1> r(t) = \lt e^t \sin t, e^t \cos t, 1\gt


  • 6.
    Finding the Binormal Vector
    Find the binormal vector for the given vector function:

    r(t)=<sin(t),cos(t),t> r(t) = \lt \sin (t), \cos (t), t \gt