From A ( − 1 , 1 , − 1 ) a perpendicular is drawn to plane P 1 with foot (1, -1, 1). From B ( 3 , − 1 , − 1 ) a perpendicular is drawn to plane P 2 with foot ( 1 , 2 , 1 ). Plane P 3 contains th

Question

From A ( − 1 , 1 , − 1 ) a perpendicular is drawn to plane P 1 ​ with foot (1, -1, 1). From B ( 3 , − 1 , − 1 ) a perpendicular is drawn to plane P 2 ​ with foot ( 1 , 2 , 1 ). Plane P 3 ​ contains the line P 1 ​ ∩ P 2 ​ and passes through the midpoint of A B . Which statements are correct?

TestHub · Accepted Answer

The planes are P 1 ​ : x − y + z − 3 = 0 and P 2 ​ : 2 x − 3 y − 2 z + 6 = 0 . The cross product of their normals is proportional to (5, 4, -1), giving A and C. The member P 1 ​ + λ P 2 ​ = 0 through the midpoint (1, 0,-1) gives P 3 ​ : 16 x − 19 y + 4 z = 12 . Its distance from the origin is 12/ 633 ​  = 2/ 18 ​ .

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