Sachin is designing a roof structure for a community center in New Delhi. The roof plane passes through three pillars located at (2, 2, −1), (3, 4, 2), and (7, 0, 6) in the site grid. What is the Cartesian equation of the roof plane?

Sachin is designing a roof structure for a community center in New Delhi. The roof plane passes through three pillars located at (2, 2, −1), (3, 4, 2), and (7, 0, 6) in the site grid. What is the Cartesian equation of the roof plane?

• A. 5x + 2y − 3z = 17
• B. 5x − 2y + 3z = 3
• C. 2x + y + z = 5
• D. x + y + z = 3

Answer: A) 5x + 2y − 3z = 17

Explanation: Points A(2, 2, −1), B(3, 4, 2), C(7, 0, 6). Vectors AB = (1, 2, 3), AC = (5, −2, 7). Normal n = AB × AC = i(14 − (−6)) − j(7 − 15) + k(−2 − 10) = 20i + 8j − 12k. Simplifying DRs: 5, 2, −3. Equation: 5(x−2) + 2(y−2) − 3(z+1) = 0 → 5x − 10 + 2y − 4 − 3z − 3 = 0 → 5x + 2y − 3z = 17.