Question

If `A = <4 ,1 ,8 >`, `B = <5 ,7 ,3 >` and `C=A-B`, what is the angle between A and C?

Answer 1

The angle is `=65.0^@`

Explanation:

Start by calculating

`vecC=vecA-vecB`

`vecC=〈4,1,8〉-〈5,7,3〉=〈-1,-6,5〉`

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=〈4,1,8〉.〈-1,-6,5〉=-4-6+40=30`

The modulus of `vecA`= `∥〈4,1,8〉∥=sqrt(16+1+64)=sqrt81=9`

The modulus of `vecC`= `∥〈-1,-6,5〉∥=sqrt(1+36+25)=sqrt62`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=30/(9*sqrt62)=0.42`

`theta=arccos(0.42)=65.0^@`