Question

What is the angle between `<-3,5,6>` and `<9,-1,8>`?

Answer 1

The angle is `46.2º`

Explanation:

The angle is obtained by using the definition of the dot product
`vecu.vecv=∣vecu∣*∣vecv∣costheta`
where `theta` is the angle between the vectors `(vecu,vecv)`

So `costheta=(vecu.vecv)/(∣vecu∣*∣vecv∣)`

Here `vecu=〈-3,5,6〉`

and `vecv=〈-9,-1,8〉`

The dot product is given by `vecu.vecv=〈u_1,u_2,u_3〉〈v_1,v_2,v_3〉=u_1v_1+u_2v_2+u_3v_3`
So `vecu.vecv=27-5+48=70`

`∣vecu∣=sqrt(u_1^2+u_2^2+u_3^2)=sqrt(9+25+36)=sqrt70`

`∣vecv∣=sqrt(v_1^2+v_2^2+v_3^2)=sqrt(81+1+64)=sqrt146`

So `costheta=70/(sqrt70.sqrt146)=`
`theta=46.2º`