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June 19th, 2017, 10:06 AM   #1
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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?


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June 19th, 2017, 10:46 AM   #2
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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

$\forall c \neq 0$

$z = \dfrac{d - a x - b y}{c}$
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June 21st, 2017, 10:33 AM   #3
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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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