Question

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

Answer 1

The angle is `=118.5`º

Explanation:

Let's start by calculating

`vecC=vecA-vecB`

`vecC=〈2,4,-1〉-〈-1,8,-5〉=〈3,-4,4〉`

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=〈2,4,-1〉.〈3,-4,4〉=6-16-4=-14`

The modulus of `vecA`= `∥〈2,4,-1〉∥=sqrt(4+16+1)=sqrt21`

The modulus of `vecC`= `∥〈3,-4,4〉∥=sqrt(9+16+16)=sqrt41`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=-14/(sqrt21*sqrt41)=-0.48`

`theta=118.5`º