Question

If `A = <6 ,4 ,-3 >`, `B = <7 ,-1 ,5 >` and `C=A-B`, what is the angle between A and C?

Answer 1

The angle is `=59.1`º

Explanation:

Let's start by calculating

`vecC=vecA-vecB`

`vecC=〈6,4,-3〉-〈7,-1,5〉=〈-1,5,-8〉`

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=〈6,4,-3〉.〈-1,5,-8〉=-6+20+24=38`

The modulus of `vecA`= `∥〈6,4,-3〉∥=sqrt(36+16+9)=sqrt61`

The modulus of `vecC`= `∥〈-1,5,-8〉∥=sqrt(1+25+64)=sqrt90`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=38/(sqrt61*sqrt90)=0.51`

`theta=59.1`º