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Let P1: 2x + y - z = 3 and P2: x + 2y + z = 2 be two planes. Which of the following statements is/are TRUE? (A) The line of intersection of P1 and P2 has direction ratios 1, 2, -1. (B) The line (3x-4)/9 = (1-3y)/9 = z/3 is perpendicular to the line of intersection of P1 and P2. (C) The acute angle between P1 and P2 is 60 degrees. (D) If P3 passes through the point (4, 2, -2) and is perpendicular to the line of intersection of P1 and P2, then the distance from (2, 1, 1) to P3 is 2/sqrt(3).

1. (A) The line of intersection of P1 and P2 has direction ratios 1, 2, -1.
2. (B) The line (3x-4)/9 = (1-3y)/9 = z/3 is perpendicular to the line of intersection of P1 and P2.
3. (C) The acute angle between P1 and P2 is 60 degrees.
4. (D) If P3 passes through (4, 2, -2) and is perpendicular to the line of intersection of P1 and P2, the distance from (2, 1, 1) to P3 is 2/sqrt(3).

Correct answer: (C) The acute angle between P1 and P2 is 60 degrees.

Solution

(A) n1 x n2 = i(1*1-(-1)*2) - j(2*1-(-1)*1) + k(2*2-1*1) = (3,-3,3), direction ratios (1,-1,1), not (1,2,-1). FALSE. (B) The given line has direction (3,-3,3) proportional to (1,-1,1), which is PARALLEL (not perpendicular) to the intersection line. FALSE. (C) cos(theta)=|n1.n2|/(|n1||n2|)=|2+2-1|/(sqrt(6)*sqrt(6))=3/6=1/2 => theta=60 deg. TRUE. (D) P3: 1*(x-4)-1*(y-2)+1*(z+2)=0 => x-y+z=0. Distance from (2,1,1): |2-1+1|/sqrt(3)=2/sqrt(3). TRUE. Correct statements: C and D.

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