Question

If `A = <2 ,-8 ,5 >`, `B = <4 ,-7 ,-5 >` and `C=A-B`, what is the angle between A and C?

Answer 1

The angle is `=56.9`º

Explanation:

Let's start by calculating

`vecC=vecA-vecB`

`vecC=〈2,-8,5〉-〈4,-7,-5〉=〈-2,-1,10〉`

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=〈2,-8,5〉.〈-2,-1,10〉=-4+8+50=54`

The modulus of `vecA`= `∥〈2,-8,5〉∥=sqrt(4+64+25)=sqrt93`

The modulus of `vecC`= `∥〈-2,-1,10〉∥=sqrt(4+1+100)=sqrt105`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=54/(sqrt93*sqrt105)=0.55`

`theta=56.9`º