Question

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

Answer 1

The angle is `=160.9`º

Explanation:

Let's start by calculating

`vecC=vecA-vecB`

`vecC=〈3,2,5〉-〈5,2,8〉=〈-2,0,-3〉`

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=〈3,2,5〉.〈-2,0,-3〉=-6+0-15=-21`

The modulus of `vecA`= `∥〈3,2,5〉∥=sqrt(9+4+25)=sqrt38`

The modulus of `vecC`= `∥〈-2,0,-3〉∥=sqrt(4+0+9)=sqrt13`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=-21/(sqrt38*sqrt13)=-0.945`

`theta=160.9`º