The equation of the plane passing through the points having position vectors ( a + b ), ( b + c ) and ( a + c ) is

Step-by-step solution

Let the given points be A, B, C with position vectors A = a + b , B = b + c , and C = a + c . Two vectors lying on the plane are: B - A = ( b + c ) - ( a + b ) = c - a C - A = ( a + c ) - ( a + b ) = c - b A normal vector to the plane is given by the cross product of these two vectors: n = ( B - A ) ( C - A ) n = ( c - a ) ( c - b ) n = c c - c b - a c + a b Since c c = 0 , - c b = b c , and - a c = c a , we get: n = a b + b c + c a The equation of the plane passing through A with normal n is: ( r - A ) n = 0 r n =