Example 4: Tangent Vector is Orthogonal to Vector of Constant Length

In this video I show that the tangent vector (derivative of a vector function) is orthogonal or perpendicular to a vector of constant length. Geometrically this means that the curve is on the surface of a sphere, since it has a constant distant or radius from the origin. I prove this by using the earlier Formula 4 for the derivative of a dot product of vector functions, and taking the derivative of the dot product between the vector and itself. Since the vector has a constant length, applying the geometric interpretation of the dot product equation makes one side of Formula 4 equal to zero, thus requiring that the dot product of the tangent vector with our given vector is equal to zero. And a dot product of zero means the vectors are perpendicular!

I also graph out this sphere and the tangent vector using the amazing GeoGebra 3D graphing calculator, so play around with it yourself! https://www.geogebra.org/3d/qjahrjrw

#math #vectors #calculus #geogebra #education

Timestamps:

- Example 4: Derivative of Vector Function of Vector with constant length: 0:00
- Recap on the Geometric Interpretation of the Dot Production formula: 0:40
- Dot product of the vector function with itself is equal to the constant squared 2:04
- Applying Formula 4 Derivative of Dot Product of Vector Functions: 4:07
- Dot Product of Derivative and the Vector Function is equal to zero, which means its orthogonal: 6:03
- Geometrically, this results in the tangent vector of a curve on a sphere: 7:57
- Graphing the tangent vector and sphere using GeoGebra (and Grok AI for the formula): 9:31
- I added another tangent vector, which is the sum of the vertical and horizontal tangent vectors: 11:32

Notes and playlists:

- Summary: SOON
- Notes: https://peakd.com/hive-128780/@mes/hfsetrue
- Sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FgQ3jBd9IFaWOakOWbHous
- Vector Functions playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FgQ3jBd9IFaWOakOWbHous .

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