Find the reflection of the point (1, 2, −1) in the plane 3x − 5y + 4z = 5.

Find the reflection of the point (1, 2, −1) in the plane 3x − 5y + 4z = 5.

• A. (73/25, −6/5, 39/25)
• B. (4/5, −3/5, 2/5)
• C. (−1, 2, 1)
• D. (2, −1, 1)

Answer: A) (73/25, −6/5, 39/25)

Explanation: Image formula: (x−x₁)/a = (y−y₁)/b = (z−z₁)/c = −2(ax₁+by₁+cz₁−d) / (a²+b²+c²). x₁=1, y₁=2, z₁=−1. a=3, b=−5, c=4, d=5. Val = 3(1) − 5(2) + 4(−1) − 5 = 3 − 10 − 4 − 5 = −16. a²+b²+c² = 9 + 25 + 16 = 50. Ratio = −2(−16)/50 = 32/50 = 16/25. x = 1 + 3(16/25) = 73/25. y = 2 − 5(16/25) = 2 − 16/5 = −6/5. z = −1 + 4(16/25) = 39/25.