0. Theoretical Foundation
Definition: A vector is a quantity that has both magnitude and direction.
Representation: Vector AB = (x₂ - x₁, y₂ - y₁)
Triangle Law: AB + BC = AC
Unit Vector: â = a / |a|
1. Scalar Product of Vectors
a · b = |a||b|cosθ
a · b = a₁b₁ + a₂b₂
cosθ = (a₁b₁ + a₂b₂) / (|a||b|)
2. Section Formula
Internal: [(mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)]
External: [(mx₂ - nx₁)/(m-n), (my₂ - ny₁)/(m-n)]
3. Vector Geometry
Parallel: a = k · b
Perpendicular: a · b = 0
Magnitude: |a| = √(a₁² + a₂²)