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11. (a) Let u : R → R" and v : R → R". By writing u v as a summation, prove that (u · v)' = u' · v + u · v'.

11. (a) Let u : R → R" and v : R → R". By writing u v as a summation, prove that (u · v)' = u' · v + u · v'.

College Algebra
College Algebra
1st Edition
ISBN:9781938168383
Author:Jay Abramson
Publisher:Jay Abramson
Chapter6: Exponential And Logarithmic Functions
Section6.4: Graphs Of Logarithmic Functions
Problem 60SE: Prove the conjecture made in the previous exercise.
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11. (a) Let u : R → R" and v : R → R". By writing u · v as a summation, prove that (u - v)' = u' · v + u · v'.
Transcribed Image Text:11. (a) Let u : R → R" and v : R → R". By writing u · v as a summation, prove that (u - v)' = u' · v + u · v'.
(b) Let p : R → R" and v E R". Assume that v and p'(t) are orthogonal for all t E R and that p(0) is orthogonal to v. Prove that v and p(t) are orthogonal for all t e R. (c) Show that if u(t) is a unit vector for all t, then u(t) and u'(t) are orthogonal for all t. Cos 2t (d) Let p(t) = te and u(t) : Find o' and u', and hence find (yu)', leave sin 2t your expression for this derivative as the sum of two orthogonal vectors.
Transcribed Image Text:(b) Let p : R → R" and v E R". Assume that v and p'(t) are orthogonal for all t E R and that p(0) is orthogonal to v. Prove that v and p(t) are orthogonal for all t e R. (c) Show that if u(t) is a unit vector for all t, then u(t) and u'(t) are orthogonal for all t. Cos 2t (d) Let p(t) = te and u(t) : Find o' and u', and hence find (yu)', leave sin 2t your expression for this derivative as the sum of two orthogonal vectors.
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