Product Rule for Vector Derivatives | Answer Key - Edubirdie

University:
Massachusetts Institute of Technology
Course:
18.02SC | Multivariable Calculus
Academic year:

2024
Views:

279

Pages:

1
Author:

hichizabej

Product rule for vector derivatives 1. If r1 (t) and r2 (t) are two parametric curves show the product rule for derivatives holds for the dot product. Answer: This will follow from the usual product rule in single variable calculus. Lets assume the curves are in the plane. The proof would be exactly the same for curves in space. We want to prove that d(r1 · r2 ) = r�1 · r2 + r1 · r�2 . dt Let r1 = �x1 , y1 � and r2 = �x2 , y2 �. We have, r1 · r2 = x1 x2 + y1 y2 . Taking derivatives using the product rule from single variable calculus, we get d(r1 · r2 ) dt = d(x1 x2 + y1 y2 ) dt = x�1 x2 + x1 x�2 + y1� y2 + y1 y2� = (x�1 x2 + y1� y2 ) + (x1 x�2 + y1 y2� ) = �x�1 , y1� � · �x2 , y2 � + �x1 , y1 � · �x�2 , y2� � = r�1 · r2 + r1 · r�2 .

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