What are the conditions for 1^° parallelism and 2^°…

What are the conditions for $1^\circ$ parallelism and $2^\circ$ perpendicularity of the line $\vec r\times\vec p=\vec q$ and the plane $\vec r\cdot\vec N=\alpha$?

Explanation of Method

The conditions for a line and a plane in vector form to be parallel or perpendicular.

Step 1: Solution

The line has direction $\vec p$; the plane has normal $\vec N$.

	Vector form	Coordinates $\vec p=\{l,m,n\}$, $\vec N=\{A,B,C\}$
$1^\circ$ parallel	$\vec p\cdot\vec N=0$	$Al+Bm+Cn=0$
$2^\circ$ perpendicular	$\vec p\times\vec N=\vec0$	$\tfrac lA=\tfrac mB=\tfrac nC$

In case $1^\circ$ the line lies in the plane iff additionally one of its points satisfies $\vec r\cdot\vec N=\alpha$. With $\vec r_0=\tfrac{\vec p\times\vec q}{|\vec p|^2}$ this reads $(\vec p\times\vec q)\cdot\vec N=\alpha|\vec p|^2$.

Result

\[\begin{aligned} 1^\circ&\ \vec p\cdot\vec N=0\quad(Al+Bm+Cn=0)\\ 2^\circ&\ \vec p\times\vec N=\vec0\quad\bigl(\tfrac lA=\tfrac mB=\tfrac nC\bigr) \end{aligned}\]