Question

What is the angle between `<-2,5,-7 >` and `< 6,-4,5>`?

Answer 1

The angle is `87.8º`

Explanation:

Let the two vectors be `vecu=〈u_1,u_2,u_3〉`
and `vecv=〈v_1,v_2,v_3〉`

The angle betwwen the vectors is `theta`

Then by the definition of the dot product
`vecu.vecv=∣vecu∣∣vecv∣costheta`

where `∣vecu∣` and `∣vecv∣` are the magnitude ot the vectors

Therefore, `costheta=(vecu.vecv)/(∣vecu∣∣vecv∣)`

`vecu=〈-2,5,7〉`
`vecv=〈6,-4,5〉`

`vecu.vecv=u_1v_1+u_2v_2+u_3v_3=-12-20+35=3`

`∣vecu∣=sqrt(u_1^2+u_2^2+u_3^2)=sqrt(4+25+49)=sqrt78`
`∣vecv∣=sqrt(v_1^2+v_2^2+v_3^2)=sqrt(36+16+25)=sqrt77`

So `costheta=3/(sqrt78sqrt77)=3/77.5=0.039`

`theta=87.8º`