Equation of a Plane Through a point
and Perpendicular to a Vector

Below is shown a plane through point \( P(x_p,y_p,z_p) \) and perpendicular (orthogonal) to vector \( \vec n = \lt x_n,y_n,z_n \gt \).

Plane through a point and perpendicular (orthogonal) to a vector

Since \( \vec {n} \) is perpendicular to the plane, any point \( M(x,y,z) \) is on the plane if the dot product of \( \vec n = \lt x_n,y_n,z_n \gt \) and vectors \( \vec {PM} = \lt x - x_p , y - y_p , z - z_p \gt \) is equal to zero.

\( \vec n \cdot \vec {PM} = \lt x_n,y_n,z_n \gt \cdot \lt x - x_p , y - y_p , z - z_p \gt = 0 \)





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