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Vectors

A vector describes a directed displacement. Its horizontal and vertical components tell you how far to move along each axis. The vector from A to B is position B minus position A; reversing the direction changes both signs.

What this guide covers

Two-dimensional vectors, translations, position vectors, magnitude, addition, subtraction, scalar multiplication and geometric reasoning. Components below are stated in words to keep direction explicit.

Syllabus reference: 2026 Mathematics 4052, G7, page 10. These are Mentora search groupings, not new official syllabus categories.

Worked example

A = (−2, 1) and B = (4, 9). Find vector AB and its magnitude.

Subtract A from B: horizontal component 4 − (−2) = 6, vertical component 9 − 1 = 8. Its magnitude is √(6² + 8²) = 10 units. Vector BA has components −6 and −8 but the same magnitude.

Try three practice checks

Attempt each on paper before revealing the worked answer. These sample particular skills; they do not diagnose the whole topic or establish mastery.

Check 1

A translation moves (3, −1) by 4 units left and 6 units up. Find the image.

Show worked answer

Add the displacement components: (3 − 4, −1 + 6) = (−1, 5).

Check 2

Vector a has components (2, −3). Find the components of −2a.

Show worked answer

Multiply both components by −2: (−4, 6). The negative scalar reverses direction and doubles the magnitude.

Check 3

Position vectors of P and Q have components (1, 4) and (7, 2). Find vector PQ.

Show worked answer

Subtract position P from position Q: (7 − 1, 2 − 4) = (6, −2).

Mistakes to watch for

  • Vector AB and vector BA have opposite directions; their magnitudes alone do not distinguish them.
  • Multiply every component by a scalar, including negative signs.
  • Parallel vectors need not have equal lengths. A scalar-multiple relation establishes parallel direction, but collinearity also needs the points or paths to be connected appropriately.

If one of these occurs, compare the relevant step with the worked example, then try a different question before drawing conclusions.

Useful foundations to check

These connections suggest what may help with this skill. Selecting a topic says nothing about your current attainment.

Applications and next topics

Combine vector paths to reason about ratios and parallel lines. Coordinate geometry offers a second way to check lengths and positions.

Choose a useful next resource

To understand a recurring error rather than relearn the whole skill: Organise multi-step vector working.

Explore the foundations connected to vectors

Sources and review

Original Mentora public practice, authored for this guide. Worked answers and syllabus mapping checked during editorial QA on 2026-09-11; no independent expert review is claimed. Graph connections are Mentora learning hints, not an official SEAB sequence. Read the source and examination-year notes.