Question

If `A = <8 ,1 ,-5 >`, `B = <6 ,-2 ,4 >`, and `C=A-B`, what is the angle between A and C?

Answer 1

The answer is `=45.9`º

Explanation:

Let's start by calculating

`vecC=vecA-vecB`

`vecC=〈8,1,-5〉-〈6,-2,4〉=〈2,3,-9〉`

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=〈8,1,-5〉.〈2,3,-9〉=16+3+45=64`

The modulus of `vecA`= `∥〈8,1,-5〉∥=sqrt(64+1+25)=sqrt90`

The modulus of `vecC`= `∥〈2,3,-9〉∥=sqrt(4+9+81)=sqrt94`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=64/(sqrt90*sqrt94)=0.696`

`theta=45.9`º