Find the equation of the plane that passes through points...

Question

Find the equation of the plane that passes through points (0, 1, 2),(2, 1, 0), and (1, 1, 0).

Is it perpendicular to the plane 2x + 3z − 2 = 0?

Answer 1

Note how the y co-ordinate, 1, is the same for all three points. Thus, the plane is y = 1 (you can verify that the three points are not collinear by seeing that the z-coordinate is 0 for two of the points and different for the third). (You can't find the plane so easily often!)

The normal vector to this plane is (0, 1, 0)

The normal vector for the other plane is (2, 0, 3)

Clearly, their inner-product is 0, so they are perpendicular, and the planes are perpendicular.