It isn't difficult for me to imagine a plane based on three points. Also it is quite simple to imagine a plane based on point and normal vector.
Are there some tricks to imagine a plane defined by plane equation $Ax+By+Cz+D=0?$
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The tag "algebraic geometry" doesn't fit. – Martin Brandenburg Jun 13 '13 at 17:07
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You could just find 3 solutions and imagine the plane through those 3 points, or you could find one solution and take $(A, B, C)$ as the normal vector (as Matt suggests below). – Jim Jun 13 '13 at 17:20
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The vector $\mathbf{a}=(A,B,C)$ is the normal vector of the plane, and the equation says that the dot product of the normal vector $\mathbf{a}$ and all vectors from the origin to a point on the plane is a constant given by $-D$. Let $\mathbf{x}=(x,y,z)$, then we have:
$$\mathbf{a}\cdot\mathbf{x}=-D$$
If the normal vector $\mathbf{a}$ is normalized to length 1, then $D$ is the distance of the plane from the origin.
Matt L.
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