Question

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

Answer 1

The angle is `=83.5`º

Explanation:

Let's start by calculating

`vecC=vecA-vecB`

`vecC=〈4,3,-7〉-〈5,7,-4〉=〈-1,-4,-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=〈4,3,-7〉.〈-1,-4,-3〉=-4-12+21=5`

The modulus of `vecA`= `∥〈4,3,-7〉∥=sqrt(16+9+49)=sqrt74`

The modulus of `vecC`= `∥〈-1,-4,-3〉∥=sqrt(1+16+9)=sqrt26`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=5/(sqrt74*sqrt26)=0.114`

`theta=83.5`º