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Vectors

Vector coordinates, magnitude, dot product, parallelism/perpendicularity, and basic geometric applications.

Practice →Formula Book →Full mock →

lessons in this chapter

previous exam questions

3 hr

suggested study time

Hard

chapter difficulty

Last reviewed Sep 2026 · GetCSCA editors

1. Vectors

Study order

Work the lessons top to bottom — each ends with practice, a timed set and a 5-stage drill. Then prove it in a full 48Q mock.

Vectors

Lesson 1 · target ~70s per question

Vector addition and scaling, magnitude, dot product and angles, plus parallel/perpendicular tests.

Why CSCA tests this

Vectors algebraize CSCA geometry: parallelism, perpendicularity, and angles become one dot-product or proportionality check instead of a construction.

Key points — memorize

Build vectors from coordinates: AB = b − a (tip minus tail), and scale/add component-wise for linear combinations.
Magnitude is Pythagoras on components; dividing by it gives the unit vector in the same direction.
The dot product has two faces: coordinate form x₁x₂ + y₁y₂ and geometric form |a||b|cos θ — set them equal to find angles.
Perpendicular (nonzero) vectors have zero dot product; parallel vectors have proportional components (x₁y₂ = x₂y₁ in 2D).
Prove midpoints and collinearity with position vectors: the midpoint is (a + b)/2 and collinear points differ by a scalar multiple.

Formula sheet

|a| = √(x² + y²)
a·b = x₁x₂ + y₁y₂ = |a||b|cos θ
cos θ = (a·b)/(|a||b|)
a ⊥ b ⇔ a·b = 0
a ∥ b ⇔ x₁y₂ = x₂y₁ (2D)
|a|² = a·a; unit vector û = a/|a| (a ≠ 0)
Midpoint of AB: (a + b)/2; vector AB = b − a (position-vector difference)
Projection of a onto b: ((a·b)/|b|²) b; scalar projection (a·b)/|b|

Classic traps

Reversing AB as a − b instead of b − a (tip minus tail).
Concluding perpendicular from a zero component rather than a zero dot product.
Dividing by |a| = 0 when normalizing the zero vector.
Mixing the angle formula numerator/denominator: cos θ = (a·b)/(|a||b|), not the reciprocal.

Exam tactic

For any angle/perpendicular question, compute a·b first — zero ends it (90°), otherwise feed it straight into cos θ.

Worked examples

Example 1: Given a = (3, 4), find |a| and its unit vector.
1. |a| = √(9 + 16) = 5.
2. Unit vector: (3/5, 4/5).
Answer: |a| = 5; unit vector (3/5, 4/5)
Example 2: Are a = (1, 2) and b = (−2, 1) perpendicular?
1. a·b = 1·(−2) + 2·1 = 0.
Answer: Yes — dot product is 0

FoundationPractice →BasicPractice →CSCA StandardPractice →Timed set · ~70s/Q →

Previous exam questions

Real CSCA-style questions tagged to Vectors — answer right here.

3 previous exam questions · 0/0 correct

Dec 2025 · Q45vectorssource issue
P midpoint of AB: vector $OP =$
No correct option in the source paper
Source issue: correct (OA+OB)/2 absent; A and C duplicate.
Correct math
True midpoint: $OP = \frac{OA+OB}{2}$ — missing. A and C are also identical.
Full solution — 3 steps
1. Midpoint formula: $OP = \frac{OA+OB}{2}$ .
2. That exact form is missing; options A and C are also identical.
3. Source issue — but $\frac{OA+OB}{2}$ is the correct math.
Jan 2026 · Q43vectors
With $AB, BC, CD$ given: collinear triple is
Apr 2026 · Q44vectors
Square ABCD, E midpoint of CD, $AB = a$ , $AD = b$ : $BE =$

Related guides

30-day study plan → fit this chapter into a week-by-week system
75-second pacing rule → hold 75s/question once the content clicks
Top 10 mistakes → the error patterns that cost the most points

1 lessons · one chapter

Finished Vectors? Prove it.

Run a timed set, clear every mistake, then take a full 48Q mock.

Practice questions →Full mock →