Question

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

Answer 1

The angle is `=130.4`º

Explanation:

Let's start by calculating

`vecC=vecA-vecB`

`vecC=〈3,-1,5〉-〈5,2,8〉=〈-2,-3,-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,-1,5〉.〈-2,-3,-3〉=-6+3-15=-18`

The modulus of `vecA`= `∥〈3,-1,5〉∥=sqrt(9+1+25)=sqrt35`

The modulus of `vecC`= `∥〈-2,-3,-3〉∥=sqrt(4+9+9)=sqrt22`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=-18/(sqrt35*sqrt22)=-0.65`

`theta=130.4`º