Question

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

Answer 1

The angle is `=54.6^@`

Explanation:

Start by calculating

`vecC=vecA-vecB`

`vecC=〈7,-3,5〉-〈4,-7,-5〉=〈3,4,10〉`

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,-3,5〉.〈3,4,10〉=21-12+50=59`

The modulus of `vecA`= `∥〈7,-3,5〉∥=sqrt(49+9+25)=sqrt83`

The modulus of `vecC`= `∥〈3,4,10〉∥=sqrt(9+16+100)=sqrt125`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=59/(sqrt83*sqrt125)=0.58`

`theta=arccos(0.58)=54.6^@`