# 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

$\vec{u}$ and $\vec{v}$ are orthogonal

#### Explanation:

Given two vectors $\vec{u} = \left\{{u}_{1} , {u}_{2}\right\}$ and $\vec{v} = \left\{{v}_{1} , {v}_{2}\right\}$
their scalar product $\left\langle\vec{u} , \vec{v}\right\rangle$ is deffined as

$\left\langle\vec{u} , \vec{v}\right\rangle = {\sum}_{i} {u}_{i} {v}_{i} = {u}_{1} {v}_{1} + {u}_{2} {v}_{2}$

In the presented case we have

$\left\langle\vec{u} , \vec{v}\right\rangle = \cos \theta \sin \theta - \sin \theta \cos \theta = 0$

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

Of course

$\left\lVert \vec{u} \right\rVert = \sqrt{\left\langle\vec{u} , \vec{u}\right\rangle} = 1$

and also

$\left\lVert \vec{v} \right\rVert = \sqrt{\left\langle\vec{v} , \vec{v}\right\rangle} = 1$