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June 19th, 2017, 10:06 AM  #1 
Senior Member Joined: Nov 2015 From: Alabama Posts: 140 Thanks: 17 Math Focus: Geometry, Trigonometry, Calculus  Can two different equations define the same plane in space?
Hello all, I have been studying planes, and had a quick question that was on my mind. Can you have two different equations in the form of ax+by+cz=d that represent the same plane in space? For the equation of a plane we need a normal vector and a point on the plane. I suppose the normal vector for two equations that are representing the same plane, would have to be the same? However, the point could be different. Using a different point, would this impact the equation of the plane we are defining in space? After thought (after question more so), given a plane ax+by+cz=d, could you find a point on that plane given the equation that defines the plane? Essentially working backwards to find a point on the plane given the equation? Thanks, Jacob 
June 19th, 2017, 10:46 AM  #2 
Senior Member Joined: Sep 2015 From: Southern California, USA Posts: 1,411 Thanks: 716 
if you have $a x + b y + c z =d$ then $\gamma a x + \gamma b y + \gamma c z = \gamma d,~\forall \gamma \in \mathbb{R}$ represents the same plane. As far as finding a point on the plane simply choose values for $x,~y$ and then $\forall c \neq 0$ $z = \dfrac{d  a x  b y}{c}$ 
June 21st, 2017, 10:33 AM  #3 
Senior Member Joined: Oct 2013 From: New York, USA Posts: 549 Thanks: 78 
Regardless of how many variables there are, an equation can be multiplied by an infinite amount of constants to produce an infinite amount of identical equations.


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