Let theta be the angle between the vectors A=(1,1,...,1) and B=(1,2,...,n) in RR^n. Find the limiting value of theta as n to oo.

Awainaideannagi

Awainaideannagi

Answered question

2022-07-22

Let θ be the angle between the vectors A = ( 1 , 1 , . . . , 1 ) and B = ( 1 , 2 , . . . , n ) in R n . Find the limiting value of θ as b
For this question, I want to apply the equation for the angle between two vectors: θ = a r c c o s A · B A B . For A·B, I use A · B = k = 1 n a k b k . And for the norm of A and B, I simply use the definition to get A = n and B = k = 1 n k 2 . But I'm stuck here. I don't know how to calculate the limit of θ as b by these equations.

Answer & Explanation

decoratesuw

decoratesuw

Beginner2022-07-23Added 11 answers

Knowing that
{ k = 1 n k = n ( n + 1 ) 2 k = 1 n k 2 = n ( n + 1 ) ( 2 n + 1 ) 6
You get
A n B n A n B n = n ( n + 1 ) / 2 n n ( n + 1 ) ( 2 n + 1 ) / 6 3 2
as n and therefore
θ = π 6
as arccos is continuous.

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