Vector Parallelogram Law?

📌 CONCEPT: The Vector Parallelogram Law states that the sum of the squares of the magnitudes of two vectors is equal to twice the sum of their dot products.

📐 RULE / FORMULA: |A + B|^2 = |A|^2 + |B|^2 + 2|A||B|cos(θ), where θ is the angle between vectors A and B.

💡 WORKED EXAMPLE: Given vectors A = i + 2j and B = 2i + 3j, we first calculate their magnitudes: |A| = √(1^2 + 2^2) = √5, |B| = √(2^2 + 3^2) = √13. Next, we find the dot product A·B = (1)(2) + (2)(3) = 8. Then, we substitute these values into the formula to verify the parallelogram law.

⚠️ COMMON MISTAKE: Students often forget to calculate the actual angle between the vectors or misinterpret the formula, leading to incorrect conclusions.