The scalar product is defined as r*s = the sum of all r*s.
Using this definition, prove that r*(u+v) = r*u + r*v.
Also, if r and s are vectors that depend on time, prove that the product rule for differentiation applies to r*s.
Ok, so I'm new to proofs and I literally do not know where to even start. Any suggestions?
First suppose $r,u,v$ are arbitrary vectors $$ r = (r_1,\dots,r_n) \quad u = (u_1,\dots,u_n) \quad v = (v_1,\dots,v_n) $$ Now write explicity both sides of your equation $$ r \ast (u+v) = \sum_{i=1}^n \, r_i \, (u_i+v_i) =\sum_{i=1}^n \, r_i \, u_i + \sum_{i=1}^n \, r_i \, v_i = r*u + r*v $$ The same goes for the proof of the derivative, write it explicity and use leibniz rule $$ \frac{d}{dt}(u(t)*v(t))=\frac{d}{dt}\left(\sum_{i=1}^n \, u_i(t) \, v_i(t) \right) = \sum_{i=1}^n \frac{d \, u_i}{dt} v_i(t) + \sum_{i=1}^n \frac{d \, v_i}{dt} u_i(t) = $$ $$ =\frac{d \, u}{dt} * v(t) + u* \frac{d \, v}{dt} $$