Write the equation of the plane passing through the points M1 and M2, parallel to the vector

Write the equation of the plane passing through the points M1 and M2, parallel to the vector a = (1,2,1), if M1 (2,2,1), M2 (3,3,2)

Let us choose an arbitrary point A (x, y, z) in the desired plane.

Consider three vectors:

M1 A = (x – 2; y – 2; z – 1);

M1 M2 = (3 – 2; 3 – 2; 2 – 1) = (1; 1; 1);

a = (1; 2; 1).

By condition, all three vectors must be coplanar, that is, when bringing them to a common origin, the vectors lie in the same plane.

The determinant of the third order, composed of their components, in theory should be equal to zero.

∆ = (x – 2) (1 * 1 – 2 * 1) – (y – 2) (1 * 1 – 1 * 1) + (z – 1) (1 * 2 – 1 * 1) = 0.

– x + 2 + z -1 = 0;

Therefore, the desired plane:

x – z = 1.