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

1 Answer
Cesareo R.
Jul 14, 2016

vec u→u and vec v→v are orthogonal

Explanation:

Given two vectors vec u = {u_1,u_2}→u={u1,u2} and vec v = {v_1,v_2}→v={v1,v2}
their scalar product << vec u, vec v >>⟨→u,→v⟩ is deffined as

<< vec u, vec v >> = sum_i u_iv_i = u_1v_1+u_2v_2⟨→u,→v⟩=∑iuivi=u1v1+u2v2

In the presented case we have

<< vec u, vec v >> =costheta sintheta-sintheta costheta = 0⟨→u,→v⟩=cosθsinθ−sinθcosθ=0

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

Of course

norm vec u = sqrt(<< vec u, vec u >>) = 1∥∥→u∥∥=√⟨→u,→u⟩=1

and also

norm vec v = sqrt(<< vec v, vec v >>) = 1 ∥∥→v∥∥=√⟨→v,→v⟩=1

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