Question

1. Time derivatives of the cross product a) Let ?a and ?b be vectors which depend on the parameter t. By writing the cross product explicitly in terms of components, prove that the derivative of the cross product satisfies the Leibniz rule, d/dt (?a × ?b) = ?a × d?b/dt + d?a/dt × ?b. (3) b) Let ?r, ?v, and ?a be the position, velocity, and acceleration of a particle respectively. Show that d/dt (?a · (?v × ?r)) = d?a/dt · (?v × ?r). (4)

          1. Time derivatives of the cross product
a) Let ?a and ?b be vectors which depend on the parameter t. By writing the cross product explicitly in terms of components, prove that the derivative of the cross product satisfies the Leibniz rule,
d/dt (?a &#215; ?b) = ?a &#215; d?b/dt + d?a/dt &#215; ?b. (3)
b) Let ?r, ?v, and ?a be the position, velocity, and acceleration of a particle respectively. Show that
d/dt (?a &#183; (?v &#215; ?r)) = d?a/dt &#183; (?v &#215; ?r). (4)

1. Time derivatives of the cross product
a) Let ?a and ?b be vectors which depend on the parameter t. By writing the cross product explicitly in terms of components, prove that the derivative of the cross product satisfies the Leibniz rule,
d/dt (?a #215; ?b) = ?a #215; d?b/dt + d?a/dt #215; ?b. (3)
b) Let ?r, ?v, and ?a be the position, velocity, and acceleration of a particle respectively. Show that
d/dt (?a #183; (?v #215; ?r)) = d?a/dt #183; (?v #215; ?r). (4)

Added by Victor Manuel R.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Timothy James

01/23/2023

Step 1

We can do this by using the cross product theorem: da (a . (U x7)) = (6 x7). dt dt Show more…

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1. Time derivatives of the cross product a) Let a and b be vectors which depend on the parameter t. By writing the cross product explicitly in terms of components, prove that the derivative of the cross product satisfies the Leibniz rule, d/dt (a x b) = a x db/dt + da/dt x b (3) b) Let r, v, and a be the position, velocity, and acceleration of a particle respectively. Show that d/dt (a * (v x r)) = da/dt * (v x r). (4)