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Vector Algebra

Vector equations, magnitudes from lengths and angles, areas and triple products. Short questions; marks are lost to sign and order slips.

Questions in the bank
182
q/paper in 2025–26
1.19
Numeric answer
24%
Notes pages
6

Tier: Core

When you’ll see it

Vectors given by components or only by lengths and angles, with a dot, cross or triple product, a projection, or an unknown vector to find.

How this chapter is tested

Vector equations — r × a = b × a, or a × c = b with a · c given — are the most frequent single type. They look different but reduce to two facts: a cross product of zero means two vectors are parallel, and crossing again with a known vector brings out the unknown.

Magnitude questions give no components, only lengths and angles, so the only tool is |v|² = v · v, expanded. Cross-product questions are mostly areas; triple-product questions are coplanarity tests or volumes, one determinant each.

The chapter is the toolkit for three-dimensional geometry: the cross product gives a normal, and the scalar triple product gives the shortest distance between skew lines. Most questions are short, and marks are lost to sign and order slips rather than to hard ideas.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Dot product: angles and projections

    cos θ = a · b / (|a||b|); scalar projection a · b / |b|, vector projection (a · b / |b|²) b.

  • Magnitudes

    |a + b|² = |a|² + |b|² + 2a · b; for unit vectors |a × b| = sin θ.

  • Cross product: areas and normals

    Triangle ½|a × b|; parallelogram |a × b| from sides or ½|d₁ × d₂| from diagonals; |a × b|² + (a · b)² = |a|²|b|².

  • Vector equations

    (r − b) × a = 0 gives r = b + λa; for a × c = b, cross both sides with a and use the given a · c.

  • Triple products

    [a b c] is a determinant and a volume, zero for coplanar vectors; a × (b × c) = (a · c)b − (a · b)c.

  • Vectors in geometry

    Section formula, centroid and collinearity; â + b̂ lies along the angle bisector.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • Order in the cross product

    c × b = −(b × c), so a × c = c × b gives (a + b) × c = 0, not (a − b) × c = 0.

  • Not associative

    a × (b × c) and (a × b) × c are different vectors. Check which pair is inside the bracket before expanding.

  • Projection divided by the wrong power

    The scalar projection divides by |b|; the projection vector divides by |b|² and then multiplies by b.

  • Edges, not positions

    Four points are coplanar when the three edges from one of them are coplanar. Testing the four position vectors tests something else.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Vector Algebra notes

Drill every Vector Algebra question

182 questions from the bank, across 6 subtopics.

Drill the 182 questions

Drill one subtopic at a time

The 6 subtopics, in teaching order.

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