"parallel" planes?

Question

Suppose you have two flat rectangular regions in $\mathbb R ^3$ such that if you expanded each into a plane, the planes would not intersect. Would you say the regions are parallel? Or is there a better word for this?

Answer 1

Yes. A plane is uniquely determined by a point on the space and a unitary vector, which is normal to the plane. We say that two planes are parallel if they don't intersect, this is equivalent as showing that their normal vector is the same (up to change of sign) and there exists at least one point that they do not have in common.

Answer 2

Parallel is the correct term to define this situation and those region can be viewed also as subsets (i.e. contained) of two planes with parallel normal vectors.

Remember that the equation for a plane in 3d space is

$$ax+by+cz+d=0$$

and $\vec n=(a,b,c)$ is a normal vector to the plane.

Answer 3

A plane in $\mathbb{R}^3$ can be defined as

$d=ax+by+cz$ where $(a,b,c)$ is the normal vector. Remember the normal vector is perpendicular to the plane $P$.

If you have two planes, with normal vectors $(a_1,b_1,c_1)$ and $(a_2,b_2,c_2)$, then you can tell if those planes are parallel if there exists a $\lambda\in\mathbb{R}$ such that $(a_1,b_1,c_1)= \lambda(a_2,b_2,c_2)$