Question

If `A = <7 ,-5 ,-2 >`, `B = <4 ,-8 ,-3 >` and `C=A-B`, what is the angle between A and C?

Answer 1

The angle is `=84.04`º

Explanation:

Let's start by calculating

`vecC=vecA-vecB`

`vecC=〈7,-5,-2〉-〈4,-8,-3〉=〈3,3,1〉`

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=〈7,-5,-2〉.〈3,3,1〉=21-15-2=4`

The modulus of `vecA`= `∥〈7,-5,-2〉∥=sqrt(49+25+4)=sqrt78`

The modulus of `vecC`= `∥〈3,3,1〉∥=sqrt(9+9+1)=sqrt19`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=4/(sqrt78*sqrt19)=0.104`

`theta=84.04`º