Finding Plane Through P(1,1,1), Q(1,0,2), R(3,1,0)

Calculus 3 · Axiom Academy

EXAMPLE Finding a Plane Through Three Points Two edge vectors, one cross product: building the plane through P , Q , and R . Find the equation of the plane that passes through the three points P=(1,1,1) , Q=(1,0,2) , and R=(3,1,0) . Nicely done — you turned three points into a plane by building a normal with the cross product. Two edges lie in the plane: and are directions inside the plane — always subtract a common base point. The cross product is the normal: is perpendicular to both edges at once, so it is normal to the whole plane. Result: x+2y+2z=5 — and all three points check out: , , . The same recipe works for any three non-collinear points: two differences, one cross product, then point-normal form.

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