Determining if Planes are Parallel

Question

Are two planes parallel if the magnitude of the cross product of their normal vectors is equal to $0$?

$|| \vec n_1 \times \vec n_2|| = 0$

Answer 1

Yes, because\begin{align}\left\lVert\vec n_1\times\vec n_2\right\rVert=0&\iff\vec n_1\times\vec n_2=0\\&\iff\vec n_1\text{ and }\vec n_2\text{ are parallel.}\end{align}

Answer 2

Indeed:

• Two planes are parallel if, and only if, their normal vectors are parallel
• Two vectors are parallel if, and only if, their cross product is the null vector.