Question

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

Answer 1

The angle is `=68.8`º

Explanation:

Let's start by calculating

`vecC=vecA-vecB`

`vecC=〈5,-1,3〉-〈2,-2,5〉=〈3,1,-2〉`

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,-1,3〉.〈3,1,-2〉=15-1-6=8`

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

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

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=8/(sqrt35*sqrt14)=0.36`

`theta=68.8`º