Reflecting Points on a Plane?

📌 CONCEPT: Reflecting a point P in 3D space about a given plane involves finding a point P' such that the line joining P and P' is perpendicular to the plane and passes through P.

📐 RULE / FORMULA: The formula to find the reflected point P' is P' = P + 2 * (P . n) * n, where P = (x, y, z) is the given point, n = (a, b, c) is the normal vector of the plane, and P . n represents the dot product of P and n.

💡 WORKED EXAMPLE: Consider the plane 2x - 3y + 4z = 7, so its normal vector is n = (2, -3, 4). We have a point P = (1, 2, 3) and need to find its reflection P'. First, we calculate the dot product P . n = (1 * 2) + (2 * -3) + (3 * 4) = 2 - 6 + 12 = 8. Then, P' = P + 2 * (P . n) * n = (1, 2, 3) + 2 * 8 * (2, -3, 4) = (1, 2, 3) + (32, -48, 64) = (33, -46, 67).

⚠️ COMMON MISTAKE: Students often forget to multiply the normal vector by the scalar 2 * (P . n) correctly, which leads to incorrect coordinates of the reflected point.