Question

What is the angle between `<4,8,2 >` and `< 0,-7,1>`?

Answer 1

The angle is `=146.4`º

Explanation:

You have 2 vectors, `vecu` and `vecv`

The angle is given by the dot product definition

`vecu.vecv=∥vecu∥*∥vecv∥costheta`

where `theta=` angle between the 2 vectors.

If `vecu=〈u_1,u_2,u_3〉` and `vecv=〈v_1,v_2,v_3〉`

The dot product is

`vecu.vecv=〈u_1,u_2,u_3〉.〈v_1,v_2,v_3〉`

`=u_1v_1+u_2v_2+u_3v_3`

Here, the dot product is

`〈4,8,2〉.〈0,-7,1〉=4*0-7*8+2*1=-54`

The modulus of a vector is `∥vecu∥=sqrt(u_1^2+u_2^2+u_3^2)`

The modulus of `vecu` is `∥vecu∥=∥〈4,8,2〉∥=sqrt(16+64+4)=sqrt84`

The modulus of `vecv` is `∥vecu∥=∥〈0,-7,1〉∥=sqrt(0+49+1)=sqrt50`

So,

`costheta=(vecu.vecv)/(∥vecu∥*∥vecv∥)=-54/(sqrt84*sqrt50)=-0.83`

`theta=146.4`º