I have a big hole in my math skills in that I don't really no where to begin in proving properties. I can use the dot product to work out things just not sure how to prove the properties.

Let U = (Ux, Uy, Uz), Let V = (Vx, Vy, Vz), Let W = (Wx, Wy, Wz), Let c and k be scalars. Prove the following dot properties.

A. U . V = V . U

B. U . (V + W) = U . V + U . W

C. k (U . V) = (kU) . V = U . (kV)

D. V . V = || V ||^2

E. 0 . V = 0

I have worked out D as:-

V . V = Vx Vx + Vy Vy + Vz Vz

= Vx^2 + Vy^2 + Vz^2

= ( SQRT( Vx^2 + Vy^2 + Vz^2) )^2

= ( ||V|| )^2

With A I know that,

U . V = Ux Vx + Ux Vy + Uz Vz

V . U = Vx Ux + Vy Uy + Vz Uz

I just don't know how to prove U . V = V . U without using values. My problem is the same for the remaining questions.

What topic do I need to study further to improve this skill I'm lacking?