Question

If `A = <-5 ,2 ,1 >`, `B = <-8 ,4 ,2 >` and `C=A-B`, what is the angle between A and C?

Answer 1

The angle is `=167.4º`

Explanation:

Let's start by calculating

`vecC=vecA-vecB`

`vecC=〈-5,2,1〉-〈-8,4,2〉=〈3,-2,-1〉`

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=〈-5,2,1〉.〈3,-2,-1〉=-15-4-1=-20`

The modulus of `vecA`= `∥〈-5,2,1〉∥=sqrt(25+4+1)=sqrt30`

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

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=-20/(sqrt30*sqrt14)=-0.98`

`theta=167.4`º