Question

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

Answer 1

The angle is `=107.5`º

Explanation:

Let's start by calculating

`vecC=vecA-vecB`

`vecC=〈5,2,8〉-〈2,5,3〉=〈3,-3,5〉`

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=〈5,2,8〉.〈3,3,-5〉=15+6-40=-19`

The modulus of `vecA`= `∥〈5,2,8〉∥=sqrt(25+4+64)=sqrt93`

The modulus of `vecC`= `∥〈3,-3,5〉∥=sqrt(9+9+25)=sqrt43`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=-19/(sqrt93*sqrt43)=-0.3`

`theta=107.5`º