How Do You Continue Expanding and Simplifying the Dot Product (2u+v)⋅(u-2v)?

[PiRsq]

How do you expand and simplify this one?


(2u+v).(u-2v)
=2u.u+2u.(-2v)+v.u+v.(-2v)

Where u and v are vectors and the "." is the dot. I did some but after this I am lost, how can I continue?

 

[HallsofIvy]

I don't see what your problem is. You have done
(2u+v).(u-2v)=2u.u+2u.(-2v)+v.u+v.(-2v) correctly. Now just
combine 2u.(-2v)= -4 u.v and v.u which is also u.v:
This is 2(u.u)- 3 u.v- 2(v.v)

You can also use the fact that |u|= sqrt(u.u) so that
u.u= |u|² and v.v= |v|² so that
(2u+v).(u-2v)= |u|²- 3u.v- 2|v|².

 

[PiRsq]

Got it, thanks