Position Vector of a Diagonal

📌 CONCEPT: The position vector of a point is a vector that represents the direction and magnitude of the line segment joining the origin to the point in a coordinate system.

📐 RULE / FORMULA: To find the position vector of a point D dividing a line segment externally in the ratio m:n, we use the formula Φd = ((mΦb - nΦa) / (m - n)) or Φd = ((-mΦb + nΦa) / (m - n)), where Φa and Φb are the position vectors of the endpoints of the line segment.

💡 WORKED EXAMPLE: Let's find the position vector of point D where AD:DB = 2:3 in triangle ABC. Given Φa = 2i + 3j, Φb = i - 4j, and Φc = 5i + 2j. Using the formula Φd = ((-2Φb + 3Φa) / (2 + 3)) = ((-2i + 8j + 3i + 9j) / 5) = (i + 17j) / 5 = (1/5)i + (17/5)j.

⚠️ COMMON MISTAKE: Students often confuse the formula for internal and external division of a line segment, resulting in incorrect position vectors.