Given 3 non-null vectors v,u,w and angles a=(u,v),b=(u,w),c=(v,w). Prove that &#x2212;<!-- − -->

Jordon Haley

Jordon Haley

Answered question

2022-05-12

Given 3 non-null vectors v,u,w and angles a=(u,v),b=(u,w),c=(v,w). Prove that 3 / 2 cos a + cos b + cos c 3?

Answer & Explanation

notemilyu1208

notemilyu1208

Beginner2022-05-13Added 20 answers

The picture is basically dividing a circle into 3 sectors and a,b,c are the angles of each sector.
Since, c = 2 π a b, cos c = cos ( a + b ) and
cos a + cos b + cos c = cos a + cos b + cos a cos b sin a sin b
writing x = cos a , y = cos b, and sin a = 1 x 2 , etc. (We can always choose + by choosing the angles to be π) We have,
x + y + x y 1 x 2 1 y 2 , x , y [ 1 , 1 ]
By A.M. > G.M.,
1 x 2 1 y 2 2 x 2 y 2 2 = 1 + x 2 + y 2 2
Thus,
x + y + x y 1 x 2 1 y 2 x + y + x y 1 + x 2 + y 2 2 = 1 2 ( x + y + 1 ) 2 3 2 3 2 .
And equality holds if and only if x = y = 1 / 2, which means the angles are all 2 π 3
Karissa Sosa

Karissa Sosa

Beginner2022-05-14Added 2 answers

Hint: Either we have a + b = c(or permutation) or a + b = 2 π c

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