Question #c976d

1 Answer
Aug 26, 2016

Answer:

See below

Explanation:

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