Question

If `A = <4 ,2 ,-5 >`, `B = <6 ,5 ,9 >` and `C=A-B`, what is the angle between A and C?

Answer 1

The angle is `=54.7`º

Explanation:

Let's start by calculating

`vecC=vecA-vecB`

`vecC=〈4,2,-5〉-〈6,5,9〉=〈-2,-3,-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=〈4,2,-5〉.〈-2,-3,-14〉=-8-6+70=56`

The modulus of `vecA`= `∥〈4,2,-5〉∥=sqrt(16+4+25)=sqrt45`

The modulus of `vecC`= `∥〈-2,-3,-14〉∥=sqrt(4+9+196)=sqrt209`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=56/(sqrt45*sqrt209)=0.58`

`theta=54.7`º