Question

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

Answer 1

The angle is `=36.1^@`

Explanation:

Start by calculating

`vecC=vecA-vecB`

`vecC=〈7,-5,4〉-〈4,7,-3〉=〈3,-12,7〉`

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,-5,4〉.〈3,-12,7〉=21+60+28=109`

The modulus of `vecA`= `∥〈7,-5,4〉∥=sqrt(49+25+16)=sqrt90`

The modulus of `vecC`= `∥〈3,-12,7〉∥=sqrt(9+144+49)=sqrt202`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=109/(sqrt90*sqrt202)=0.81`

`theta=arccos(0.81)=36.1^@`