Vector Position and Reflection?

📌 CONCEPT: The reflection of a point B over another point C in a Cartesian plane results in a new point B' that is equidistant from C as B is.

📐 RULE / FORMULA: The vector representing the reflected point B' can be obtained by using the formula ℓB' = 2ℓC - ℓB, where ℓC is the position vector of point C.

💡 WORKED EXAMPLE: Given A(-5, 4), C(1, -3), and ℓB = ℓ(5, -6), we first find ℓC = ℓ(1, -3). Then, using the formula, ℓB' = 2ℓ(1, -3) - ℓ(5, -6) = ℓ(7, 6). Therefore, the vector ℓB' representing vector AB' is ℓ(7, 6).

⚠️ COMMON MISTAKE: Students often incorrectly assume that the reflected point B' coincides with B + 2ℓB, but this is not true because the reflection is not a simple translation of the point B by twice the vector ℓB.