A plane passes through the points A (1, 2, 3) B (2, 3, 1) and C(2, 4, 2). If O is the origin and P is   (2, -1, 1), then the projection of OP→on this plane is of length

The given points are A (1, 2, 3) B (2, 3, 1) and C(2, 4, 2).             Equation of the plane passing through the above points is x−1y−2z−311−212−1=0            Expand the above determinant,                                                 (x−1)  (3)−(y−2)(1)+(z−3)(1)=03x−y+z−4=0            Projection of O (0,0,0) on the above plane is (h,k,l)                                  h−03=k−0−1=l−01=−(−4)9+1+1=411                           Hence, the projection of the origin on the plane is o'  =   1211,−411,411            Projection of P (2,-1,1) on the plane is (h,k,l)                                       h−23=k+1−1 =l−1l=−(6+1+1−4)9+1+1=−411                      Hence the projection of  on the plane is P'=1011,−711,711            Therefore,O'P'=4121+9121+9121=22121=211