Definition of a vector
A vector AB is defined by start point A and end point B. It has direction, sense, and magnitude.
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Lesson overview
This lesson brings together the PDF, study notes and chapter flow so you can revise Vector Calculus without losing the thread.
Definition of a vector
A vector AB is defined by start point A and end point B. It has direction, sense, and magnitude.
Coordinates
If A(xA, yA) and B(xB, yB), then AB = (xB − xA, yB − yA).
Magnitude (length)
‖AB‖ = √((xB−xA)² + (yB−yA)²)
Addition
u + v = (ux + vx, uy + vy). Parallelogram rule or Chasles rule.
Chasles relation
AB + BC = AC for any points A, B, C.
Scalar multiplication
k·u = (k·ux, k·uy). If k > 0 same direction, k < 0 opposite direction.
Collinear vectors
u and v are collinear ⟺ ux·vy − uy·vx = 0 (zero determinant).
Basis of the plane
Any vector decomposes uniquely in a non-collinear basis (i, j): u = x·i + y·j.
Problem-solving method
Identify data, set formulas, solve step by step, then verify sign, unit, and coherence.
Common mistakes
Forgetting domain constraints, mixing equality/equivalence, and losing solutions during transformations.