Given a plane, how can I tell if it's perpendicular or parallel to one of these vectors? I have this plane $z=\tan \theta (x-1)$ and a set of orthonormal basis $(\cos \theta, 0, \sin \theta), (0, 1, 0), (\sin \theta, 0, -\cos \theta)$, say $e_1, e_2, e_3$ correspondingly, how can I determine which vector is perpendicular/parallel to the plane?
 A: Let the given orthonormal basis are $ \hat e_1,\hat e_2 , \hat e_3$, then any vector $ \vec v$ can be written as $$ \begin{align} \vec v & = x \hat e_1 + y\hat e_2 + z  \hat e_3 \\
& = x(\cos \theta \hat i+ \sin \theta \hat k)+y \hat j + z( \sin \theta \hat i - \cos \theta \hat k)  \\ & = ( x \cos \theta + z \sin \theta) \hat i + y  \hat j + (x \sin \theta - z \cos \theta) \hat k \end{align} $$
If I define $ \vec n$ to be the normal of plane, then $$ \vec n = - \tan \theta \hat i + \hat k$$
Now, we have two vectors in the same orthonormal basis, it can be checked whether they are perpendicular or parallel by dot and cross products respectively.
A: If a plane $\ P\ $ in 3-dimensional space is defined by an equation of the form $\ ax + by + cz = d\ $, where $\ a,b,c\ $ are not all zero, then a vector $\ (v_1, v_2, v_3)\ $ is perpendicular to $\ P\ $ if and only if it is a scalar multiple of  $\ (a, b, c)\ $ (i.e. $\ v_1=\lambda a\ $, $\ v_2=\lambda b\ $, and $\ v_3=\lambda c\ $, for some scalar $\ \lambda\ $).  An easy way of checking this is to calculate the vector cross product $\ (v_1, v_2, v_3)\times (a, b, c)= (v_2 c-v_3 b, v_3 a - v_1 c, v_1 b - v_2 a)\ $, which will be zero if and only if it's true.
The vector $\ (v_1, v_2, v_3)\ $ is parallel to $\ P\ $ if and only if the dot product $\ (v_1, v_2, v_3)\cdot (a, b, c)= \ v_1 a + v_2 b + v_3 c\ $ is zero.
In your case, the plane has equation $\ \tan\theta x + 0 y  - z = \tan\theta\ $, so you can take $ a = \tan\theta\ $, $\ b=0\ $, and $\ c=-1\ $, and you can check which of your vectors $\ e_1\ $, $\ e_2\ $, $\ e_3\ $ are parallel or perpendicular to the plane by taking their cross and dot products with the vector $\  (\tan\theta, 0, -1)\ $.
