Question

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

Answer 1

The angle is `=42.7^@`

Explanation:

Let's start by calculating

`vecC=vecA-vecB`

`vecC=〈4,-5,1〉-〈5,8,-3〉=〈-1,-13,4〉`

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,-5,1〉.〈-1,-13,4〉=-4+65+4=65`

The modulus of `vecA`= `∥〈4,-5,1〉∥=sqrt(16+25+1)=sqrt42`

The modulus of `vecC`= `∥〈-1,-13,4〉∥=sqrt(1+169+16)=sqrt186`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=65/(sqrt42*sqrt186)=0.74`

`theta=42.7^@`