Question

If `A = <7 ,9 ,4 >`, `B = <4 ,6 ,-3 >` and `C=A-B`, what is the angle between A and C?

Answer 1

The angle is `=39.8^@`

Explanation:

Let's start by calculating

`vecC=vecA-vecB`

`vecC=〈7,9,4〉-〈4,6,-3〉=〈3,3,7〉`

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=〈7,9,4〉.〈3,3,7〉=21+27+28=76`

The modulus of `vecA`= `∥〈7,9,4〉∥=sqrt(49+81+16)=sqrt146`

The modulus of `vecC`= `∥〈3,3,7〉∥=sqrt(9+9+49)=sqrt67`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=76/(sqrt146*sqrt67)=0.77`

`theta=39.8^@`