find if 2 planes are parallel or not

Question

I have 2 parallel planes with equation :

a.x + b.y + c.z +d = 0 a'.x + b'.y + c'.z +d = 0

And I need to demonstrate they are parallel.

I could read that they are parallel if they have "2 collinear normal vectors" But I couldn't understand what it means? Well, collinear mean parallel, but does it mean I need minimu 2 true sentences on these ones ?

a=k.a' b=k.b' c=k.c'

Excuse me if this is an easy issue, but just starting to learn back mathematics...

Answer 1

With the equation of the plane in standard form $ax+by+cz = d, (a,b,c)$ is the normal vector to the plane.

If two planes are parallel then these normal vectors are scalar multiples of one another.

So, if $(a,b,c) = (ka',kb',kc')$ the normal vectors are parallel. However, if also $d = kd'$ then they are the same plane.