Question

If `A = <3 ,-1 ,8 >`, `B = <4 ,-3 ,-6 >` and `C=A-B`, what is the angle between A and C?

Answer 1

The angle is `=28.7º`

Explanation:

Let's start by calculating

`vecC=vecA-vecB`

`vecC=〈3,-1,8〉-〈4,-3,-6〉=〈-1,2,14〉`

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,-1,8〉.〈-1,2,14〉=-3-2+112=107`

The modulus of `vecA`= `∥〈3,-1,8〉∥=sqrt(9+1+64)=sqrt74`

The modulus of `vecC`= `∥〈-1,2,14〉∥=sqrt(1+4+196)=sqrt201`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=107/(sqrt74*sqrt201)=0.88`

`theta=28.7`º