Question

If `A = <3 ,1 ,-1 >`, `B = <4 ,-7 ,-1 >` and `C=A-B`, what is the angle between A and C?

Answer 1

The answer is `79.2º`

Explanation:

Lets start by calculating

`vecC=vecA-vecB=〈3,1,-1〉-〈4,-7,-1〉=〈-1,8,0〉`

The angle `theta` between `vecA` an `vecC` is given by the dot product definition

`vecA.vecC=∥vecA∥*vecC∥*cos theta`

The dot product `=〈3,1,-1〉.〈-1,8,0〉=(3*-1+1*8+0)=5`

The modulus of `vecA` is `=∥vecA∥=∥〈3,1,-1〉∥=sqrt(9+1+1)=sqrt11`

The modulus of `vecC` is `=∥vecC∥=∥〈-1,8,0〉∥=sqrt(1+64+0)=sqrt65`

The angle is `costheta=(vecA.vecC)/(∥vecA∥*vecC∥)`

`costheta=5/(sqrt11*sqrt65)=0.187`

`theta=79.2º`