Question

If `A = <2 ,1 ,8 >`, `B = <5 ,7 ,3 >` and `C=A-B`, what is the angle between A and C?

Answer 1

THe angle is 66.2º

Explanation:

The dot product betwwen 2 vectors is
`〈veca〉.〈vecc〉=∥〈veca〉∥∥〈vecc〉∥cos(veca,vecc)`

`cos(veca,vecc)=(〈veca〉 .〈 vecc 〉 ) /(∥〈veca〉∥∥〈vecc〉∥)`

If `〈veca〉=〈a_1,a_2,a_3〉`
and `〈vecc〉=〈c_1,c_2,c_3〉`

The dot product is `〈veca〉.〈vecc〉=a_1c_1+a_2c_2+a_3c_3`

`〈vecc〉=〈veca〉-〈vecb〉 =〈2,1,8〉-〈5,7,3〉 =〈-3,-6,5〉`

`〈veca〉.〈vecc〉=-6+(-6)+40=28`

`∥〈veca〉∥=sqrt(a_1^2+a_2^2+a_3^2)`

`∥〈veca〉∥=sqrt(4+1+64)=sqrt69`

`∥〈vecc〉∥=sqrt(9+36+25)=sqrt70`
let `cos(veca,vecc)=costheta`
so the angle
`costheta=28/((sqrt69)(sqrt70))=0.403`
`theta =66.2º`