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

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Answer 1 · 2016-07-14T11:15:46

$vec u$ and $vec v$ are orthogonal

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

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

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