If M denotes the midpoint of the line joining A (4, 5, -10) and B (-1, 2, 1) , then the equation of the plane through M and perpendicular to AB is:

Step-by-step solution

The coordinates of the midpoint M of the line segment joining A(4, 5, -10) and B(-1, 2, 1) are given by: M = ( 4 - 1 2 , 5 + 2 2 , -10 + 1 2 ) = ( 3 2 , 7 2 , - 9 2 ) The normal vector n to the plane is parallel to the vector AB since the plane is perpendicular to AB . n = AB = (-1 - 4) i + (2 - 5) j + (1 - (-10)) k = -5 i - 3 j + 11 k The equation of the plane passing through M with normal vector n is: ( r - m ) n = 0 r n = m n Substituting the values of m and n : m n = ( 3 2 i + 7 2 j - 9 2 k ) (-5 i - 3 j + 11 k