equation of plane

Question

The general equation of a plane in 3D is

RankGuru · Accepted Answer

1. **Definition**: A plane is a surface such that if any two points are taken on it, the entire line segment joining them lies on the surface.
2. **Normal Vector**: Any plane is uniquely determined by a point on it and a vector $\vec{n} = A\hat{i} + B\hat{j} + C\hat{k}$ normal to it.
3. **Derivation**: For a point $(x, y, z)$ on the plane passing through $(x_0, y_0, z_0)$:
   $A(x-x_0) + B(y-y_0) + C(z-z_0) = 0 \implies Ax + By + Cz - (Ax_0 + By_0 + Cz_0) = 0$
4. **General Form**: Letting $D = -(Ax_0 + By_0 + Cz_0)$, we get $Ax + By + Cz + D = 0$.

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