Question

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

Answer 1

`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`