Using a specific example, prove the vector property: For vectors u⃗ and v⃗ and scalar p > 1, show p(u⃗ +v⃗ )=pu⃗ +pv⃗ . It works for all p-values, but I am asking you to narrow your examples to a semi-specific example.
(The squares mean its a vector)
well, in two dimensions (same for any number), constants a,b,c,d
u = a i + b j
v = c i + d j
u + v = (a+c)i + (b+d)j
p[u + v] = (pa+pc)i + (pb+pd)j
now
p u = p a i + p b j
p v = p c i + p d j
so
pu + pv = (pa+pc)i + (pb+pd)j same