Question

If `A = <4 ,7 ,2 >`, `B = <1 ,1 ,-4 >` and `C=A-B`, what is the angle between A and C?

Answer 1

The angle is `=28º`

Explanation:

Let's start by calculating

`vecC=vecA-vecB`

`vecC=〈4,7,2〉-〈1,1,-4〉=〈3,6,6〉`

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=〈4,7,2〉.〈3,6,6〉=12+42+12=66`

The modulus of `vecA`= `∥〈4,7,2〉∥=sqrt(16+49+4)=sqrt69`

The modulus of `vecC`= `∥〈3,6,6〉∥=sqrt(9+36+36)=sqrt81=9`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=66/(sqrt69*9)=0.88`

`theta=28`º