Question

If `A = <2 ,4 ,-3 >`, `B = <3 ,1 ,-5 >` and `C=A-B`, what is the angle between A and C?

Answer 1

The angle is `=78.5º`

Explanation:

Let's calculate `vecC`

`vecC=vecA-vecB=〈2,4,-3〉-〈3,1,-5〉=〈-1,3.2〉`

The angle between `vecA` and `vecC` is obtained from the dot product definition

`vecA.vecC=∥vecA ∥* ∥vecC∥* costheta`

The dot product is `=〈-1,3.2〉.〈2,4,-3〉=(-2+12-6)=4`

The modulus of `vecA` is `=∥vecA∥=∥〈2,4,-3〉∥=sqrt(4+16+9)=sqrt29`

The modulus of `vecC` is `=∥vecc∥=∥〈-1,3,2〉∥=sqrt(1+9+4)=sqrt14`

Therefore,

`costheta=(vecA.vecC)/(∥vecA ∥* ∥vecC∥)=4/(sqrt29sqrt14)=0.1985`

`theta=78.5º`