Question

If `A = <3 ,-1 ,-7 >`, `B = <5 ,6 ,-4 >` and `C=A-B`, what is the angle between A and C?

Answer 1

The angle is `=68.7`º

Explanation:

Let's start by calculating

`vecC=vecA-vecB`

`vecC=〈3,-1,-7〉-〈5,6,-4〉=〈-2,-7,-3〉`

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,-7〉.〈-2,-7,-3〉=-6+7+21=22`

The modulus of `vecA`= `∥〈3,-1,-7〉∥=sqrt(9+1+49)=sqrt59`

The modulus of `vecC`= `∥〈-2,-7,-3〉∥=sqrt(4+49+9)=sqrt62`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=22/(sqrt59*sqrt62)=0.36`

`theta=68.7`º