A plane passes through (2,1,2) and (1,2,1) and parallel to the line 2 x=3 y and z =1 , then the plane also passes through the point.

Step-by-step solution

The plane contains points P₁(2,1,2) and P₂(1,2,1) , and is parallel to the line defined by 2x=3y and z=1 . A vector in the plane is given by v ₁ = P₂ - P₁ = (-1, 1, -1) . The direction vector of the line is d = (3, 2, 0) , obtained by parameterizing x = 3t , y = 2t , z = 1 from the line equations. The normal vector to the plane is perpendicular to both v ₁ and d , computed as their cross product: n = v ₁ d = vmatrix i & j & k -1 & 1 & -1 3 & 2 & 0 vmatrix = (2, -3, -5) . Using point P₁(2,1,2) and normal vector (2,