- #1
Zheng Tien
- 15
- 2
(I am not very sure if this is a high-school level question or a undergraduate level question. Sorry.)
Does our normal differentiation rules, like the product rule and quotient rule apply to vectors?
Say for example, differentiate ##r \times \dot r##
##r## is radius vector, ##\dot r## is the time derivative of the radius vector (i.e. velocity vector), and you cross both vector (thus the product is a vector, not a scalar as in dot multiplication.)
Now in normal case, assume that ##r## and ## \dot r## are both scalar, not vector, we will apply product rule, i.e. differentiate the first variable (##r##), retain the second (##r##), multiply both, plus differentiate the second variable (##r##) and add with the first variable (##r##)
So, your result is:
(##r \times \ddot r ##) + (## \dot r \times \dot r##)
But what about vectors? If both ##r## and ## \dot r## are vectors and you cross multiply or dot multiply them, will you differentiate them the same way like you do as if they were scalars?
Does our normal differentiation rules, like the product rule and quotient rule apply to vectors?
Say for example, differentiate ##r \times \dot r##
##r## is radius vector, ##\dot r## is the time derivative of the radius vector (i.e. velocity vector), and you cross both vector (thus the product is a vector, not a scalar as in dot multiplication.)
Now in normal case, assume that ##r## and ## \dot r## are both scalar, not vector, we will apply product rule, i.e. differentiate the first variable (##r##), retain the second (##r##), multiply both, plus differentiate the second variable (##r##) and add with the first variable (##r##)
So, your result is:
(##r \times \ddot r ##) + (## \dot r \times \dot r##)
But what about vectors? If both ##r## and ## \dot r## are vectors and you cross multiply or dot multiply them, will you differentiate them the same way like you do as if they were scalars?