Vectors

Vectors: direction, magnitude and proof with components

A vector describes displacement: both size and direction matter. Write a vector in bold, such as a, or as a column vector, such as (3, -2). Do not confuse it with a scalar length. Vector methods are powerful because they preserve direction and can prove geometric relationships without measuring a diagram.

Conceptual model

A two-dimensional vector can be written as a column vector (a, b). Addition is component by component: (a, b) + (c, d) = (a + c, b + d). Scalar multiplication changes size and may reverse direction: k(a, b) = (ka, kb). If position vectors of A and B from an origin O are a and b, then the vector from A to B is AB = b - a. The magnitude of (a, b) is √(a² + b²). Parallel vectors are scalar multiples; equal vectors have the same components, irrespective of where they are drawn. A line through position vector p in direction d has vector equation r = p + λd, where λ is a real parameter.

Worked or evidence example 1

Let OA = (2, -1) and OB = (7, 3). Find AB. Use endpoint minus start point: AB = OB - OA = (7, 3) - (2, -1) = (7 - 2, 3 - (-1)) = (5, 4). Its magnitude is |AB| = √(5² + 4²) = √41. The minus negative is essential. Check direction: B is five units right and four units up from A, agreeing with the component vector. The exact answer is √41, not 6.4 unless an approximation is requested.

Worked or evidence example 2

In triangle ABC, let AB = u and AC = v. Point M is the midpoint of BC. Since OB = OA + u and OC = OA + v, OM = (OB + OC)/2 = [OA + u + OA + v]/2 = OA + (1/2)u + (1/2)v. Therefore AM = OM - OA = (1/2)(u + v). This proves that the median from A has that vector. The reasoning uses position vectors and the midpoint definition; it does not depend on the triangle looking symmetric.

Answer method

Name a consistent origin and write every required route as an addition or subtraction of directed vectors. For an unknown point, translate the wording into a ratio before calculating: if AP:PB = 2:1, then AP = (2/3)AB and OP = OA + (2/3)AB. For proof, end with the vector fact and its geometric meaning: showing CD = kAB proves CD is parallel to AB; if k is positive it has the same direction, if negative the opposite direction. For intersections, equate two vector equations and solve the parameters component by component, then verify both coordinates.

Common errors and limitations

Do not write AB = OA - OB; the order is end minus start. Do not add magnitudes in place of vectors unless the vectors are known to be collinear in the relevant direction. A scalar multiple of zero needs care, and two vectors of equal magnitude need not be parallel. Keep vector arrows or bold notation distinct from lengths such as |u|. Do not assume a point is a midpoint because it appears central in a sketch; show that its position vector is the average of the endpoints or that the two directed segments are equal.

Active-recall and application drill

Given OA = (-1, 2), OB = (4, 8), calculate AB and |AB| exactly. If C divides AB internally in the ratio AC:CB = 1:2, find OC using OC = OA + (1/3)AB. Then write a two-line proof: if PQ = -3RS, what does this show about the directions and parallelism of PQ and RS? Finally, state the role of λ in r = p + λd.

Summary

Vectors encode directed movement. Use endpoint minus start, component arithmetic and scalar multiples exactly; then translate the final vector relationship into a statement about length, direction, parallelism, ratios or intersection.

Final self-check

Before writing, underline the command word. If the task asks explain, make a chain from cause to process to outcome; if it asks assess or evaluate, compare factors and give a qualified judgement. For a data source, identify the pattern that is actually visible before suggesting a cause. The pattern is evidence; the proposed cause is an inference and may need a limitation. For a calculation, show the formula or definition, substitute values with units, simplify carefully, and check whether the result fits the original condition.

Use this short audit after every answer: have I answered the precise question; have I used a relevant detail or completed line of working; have I explained its significance; and have I stated a limit where the evidence cannot prove too much? Avoid impressive-sounding generalisations. A small, accurate comparison is stronger than a broad claim with no support. Close your notes and reconstruct the model in four sentences, then revisit the question a day later. Yeh retrieval practice gaps ko jaldi expose karti hai.

When revising, change one condition in a worked example and predict what changes before calculating or writing. In a human-geography or economics answer, ask which group might experience the outcome differently and what evidence could test that claim. In mathematics, change one coordinate, vector component or parameter and identify which step must be recalculated. This variation stops memorised procedures from becoming blind routines and makes the underlying relationship clearer.