Question

If `A = <2 ,1 ,-3 >`, `B = <3 ,4 ,1 >` and `C=A-B`, what is the angle between A and C?

Answer 1

The angle is `=68.48^@`

Explanation:

Start by calculating

`vecC=vecA-vecB`

`vecC=〈2,1,-3〉-〈3,4,1〉=〈-1,-3,-4〉`

The angle between `vecA` and `vecC` is given by the dot product definition.

`vecA.vecC=∥vecA∥*∥vecC∥costheta`

Where `theta` is the angle between `vecA` and `vecC`

The dot product is

`vecA.vecC=〈2,1,-3〉.〈-1,-3,-4〉=-2-3+12=7`

The modulus of `vecA`= `∥〈2,1,-3〉∥=sqrt(4+1+9)=sqrt14`

The modulus of `vecC`= `∥〈-1,-3,-4〉∥=sqrt(1+9+16)=sqrt26`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=7/(sqrt14*sqrt26)=0.3669`

`theta=arccos(0.3669)=68.48^@`