Consider three planes P 1 : x − y + z = 1 , P 2 : x + y − z = − 1 and P 3 : x − 3 y + 3 z = 2 Let L 1 , L 2 and L 3 be the lines of intersection of the planes P 2 and P 3 , P 3 and P

Question

Consider three planes P 1 ​ : x − y + z = 1 , P 2 ​ : x + y − z = − 1 and P 3 ​ : x − 3 y + 3 z = 2 Let L 1 ​ , L 2 ​ and L 3 ​ be the lines of intersection of the planes P 2 ​ and P 3 ​ , P 3 ​ and P 1 ​ , P 1 ​ and P 2 ​ , respectively. Statement 1 Atleast two of the lines L 1 ​ , L 2 ​ and L 3 ​ are non-parallel. Statement 2 The three planes do not have a common point.

TestHub · Accepted Answer

The given equations are ​ x − y + z = 1 , x + y − z = − 1 and x − 3 y + 3 z = 2 ​ The system of equations can be put in matrix form as A X = B which is inconsistent as ρ ( A : B )  = ρ ( A ) . ⇒ The three planes do not have a common point. ⇒ Statement 2 is true. Since, planes P 1 ​ , P 2 ​ and P 3 ​ are pairwise intersection, their lines of intersection are parallel. Statement 1 is false.

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