Question

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

Answer 1

The angle is `=34.3`º

Explanation:

Let's start by calculating

`vecC=vecA-vecB`

`vecC=〈-7,1,5〉-〈3,6,1〉=〈-10,-5,4〉`

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=〈-7,1,5〉.〈-10,-5,4〉=70-5+20=85`

The modulus of `vecA`= `∥〈-7,1,5〉∥=sqrt(49+1+25)=sqrt75`

The modulus of `vecC`= `∥〈-10,-5,4〉∥=sqrt(100+25+16)=sqrt141`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=85/(sqrt75*sqrt141)=0.83`

`theta=34.3`º