Proof for Properties of Dot Product

Proof of u^→ • (v^→ + w^→) = u^→ • v^→ + u^→ • w^→

We'll begin with the three vectors, u^→ = (u₁, u₂, .... u_n),  v^→ = (v₁, v₂, ... v_n) and w^→ = (w₁,w₂,...,w_n) and yes we did mean for these to each comprise n components. The theorem works for general vectors thus we may also do the proof for general vectors.

Here now, as noted above this is pretty very much just a "computational" proof. What that means is that we'll calculate the left side and then do some basic arithmetic on the result to display that we can make the left side be like the right side. The work is here.

u^→ • (v^→ + w^→) = (u₁, u₂, .... u_n) • {( v₁, v₂, ... v_n) + (w₁,w₂,...,w_n)}

                       = (u₁, u₂, .... u_n) • (v₁ + w₁, v₂ + w₂, ... v_n + w_n)

                        = {u₁ (v₁+w₁), u₂ (v₂+w₂), .... u_n (v_n + w_n)}

                        = {u₁v₁ + u₁w₁, u₂v₂ + u₂w₂, ..., u_nv_n + u_nw_n}

                        = {u₁v₁, u₂v₂, ... , u_nv_n}+ {u₁w₁, u₂w₂, ..., u_nw_n}

                        = (u₁, u₂, .... u_n) • {( v₁, v₂, ... v_n) + (u₁, u₂, .... u_n) • (w₁,w₂,...,w_n)}

                        = u^→• v^→ + u^→ • w^→