Question

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

Answer 1

The angle is `=23.1`º

Explanation:

Let's start by calculating

`vecC=vecA-vecB`

`vecC=〈-7,1,-3〉-〈3,4,1〉=〈-10,-3,-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=〈-7,1,-3〉.〈-10,-3,-4〉=70-3+12=79`

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

The modulus of `vecC`= `∥〈-10,-3,-4〉∥=sqrt(100+9+16)=sqrt125`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=79/(sqrt59*sqrt125)=0.9199`

`theta=23.1`º