Given two vectors vec a, vec b their sum is
vec s = vec a + vec b. Now computing the norm of vec s and squaring
norm (vec s)^2 = norm( vec a + vec b)^2 = norm (vec a)^2+2 << vec a, vec b >> + norm( vec b)^2
The scalar product
<< vec a, vec b >> = norm vec a norm vec b cos(hat(vec a, vec b)) has a maximum when cos(hat(vec a, vec b))=1 and a minimum when cos(hat(vec a, vec b))=-1 so
min norm(vec s)^2 = norm (vec a)^2+norm(vec b)^2-2norm vec a norm vec b and
max norm(vec s)^2 = norm (vec a)^2+norm(vec b)^2+2norm vec a norm vec b
Finally, when two vectors vec a, vec b are aligned, their sum is a minimum or a maximum deppending on their relative orientation:
concordant a maximum
discordant a minimum
