Question

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

Answer 1

The angle is `=68.1^@`

Explanation:

Let's start by calculating

`vecC=vecA-vecB`

`vecC=〈4,-5,-4〉-〈5,1,-6〉=〈-1,-6,2〉`

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,-4〉.〈-1,-6,2〉=-4+30-8=18`

The modulus of `vecA`= `∥〈4,-5,-4〉∥=sqrt(16+25+16)=sqrt57`

The modulus of `vecC`= `∥〈-1,-6,2〉∥=sqrt(1+36+4)=sqrt41`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=18/(sqrt57*sqrt41)=0.37`

`theta=arccos(0.37)=68.1^@`