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Tangent, normal, and binormal vectors

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Intros
Lessons
  1. Tangent, Normal, & Binormal Vectors Overview:
  2. 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)||}
  3. 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)
  4. 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)
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Examples
Lessons
  1. Finding the Tangent Vector and Line
    Find the tangent line to the vector function r(t)=<t3−2,2+2t2,t>r(t)= \lt t^3-2, 2+2t^2, t\gt at t=1t=1.
    1. Find the tangent line to the vector function r(t)=<cos⁡t,sin⁡t,tan⁡t>r(t)= \lt\cos t, \sin t, \tan t\gt at t=πt= \pi.
      1. 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

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

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

          1. Find the unit normal vector for the given vector function:

            r(t)=<etsin⁡t,etcos⁡t,1> r(t) = \lt e^t \sin t, e^t \cos t, 1\gt

            1. 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

              Topic Notes
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              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)