How do you determine whether u and v are orthogonal, parallel or neither given #u=<costheta, sintheta># and #v=<sintheta, -costheta>#?

1 Answer
Jul 14, 2016

Answer:

#vec u# and #vec v# are orthogonal

Explanation:

Given two vectors #vec u = {u_1,u_2}# and #vec v = {v_1,v_2}#
their scalar product #<< vec u, vec v >># is deffined as

#<< vec u, vec v >> = sum_i u_iv_i = u_1v_1+u_2v_2#

In the presented case we have

#<< vec u, vec v >> =costheta sintheta-sintheta costheta = 0#

when this occurs with neither of #vec u, vec v# being a null vector, it is said that them are orthogonal.

Of course

#norm vec u = sqrt(<< vec u, vec u >>) = 1#

and also

#norm vec v = sqrt(<< vec v, vec v >>) = 1 #