Equation of a Plane Through three Points

Below is shown a plane passing through the three points \( P(x_p,y_p,z_p) \), \( Q(x_q,y_q,z_q) \) and \( R(x_r,y_r,z_r) \).

Plane through three points

We first define vector \( \vec {n} \) as the cross product of vectors \( \vec {PR} \) and \( \vec {PQ} \)

\( \vec {n} = \vec {PR} \times \vec {PQ} \)


From definition of the cross product, \( \vec {n} \) is perpendicular to both vectors \( \vec {PR} \) and \( \vec {PQ} \) and therefore to the plane containing all three points P, Q and R. 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.

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





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