Question

If `A = <3 ,1 ,-1 >`, `B = <4 ,6 ,-1 >` and `C=A-B`, what is the angle between A and C?

Answer 1

The angle is `=118.2`º

Explanation:

Let's start by calculating

`vecC=vecA-vecB`

`vecC=〈3,1,-1〉-〈4,6,-1〉=〈-1,-5,0〉`

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,-1〉.〈-1,-5,0〉=-3-5-0=-8`

The modulus of `vecA`= `∥〈3,1,-1〉∥=sqrt(9+1+1)=sqrt11`

The modulus of `vecC`= `∥〈-1,-5,0〉∥=sqrt(1+25+0)=sqrt26`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=-8/(sqrt11*sqrt26)=-0.47`

`theta=118.2`º