Question

If `A = <4 ,-5 ,1 >`, `B = <5 ,1 ,-3 >` and `C=A-B`, what is the angle between A and C?

Answer 1

Reqd. angle `= arccos(30/sqrt2226).`

Explanation:

`A=<4,-5,1>, B=<5,1,-3>`
`:. C=A-B= <4-5,-5-1,1-(-3)>` = `<-1,-6,4>.`

Let the reqd. angle be `theta`. Then, using dot product of `A` & `C`, we have,
`A.C=||A||*||C|\costheta rArr <4,-5,1>.<-1,-6,4>`=`sqrt(4^2+(-5)^2+1^2)*sqrt((-1)^2+(-6)^2+4^2)*costheta`

`:.-4+30+4=sqrt42*sqrt53*costheta`
`:.30=sqrt2226*costheta`
`:. costheta=30/sqrt2226`
`:. theta = arccos(30/sqrt2226).`