Vectors

For the vector from A to B, subtract the position of A from the position of B. Reversing the journey reverses both signs. This G3 Math guide works through direction, magnitude, translations and scalar multiplication before connecting them to geometric reasoning.

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.

Keep vectors and lengths separate

Position: OA describes the journey from the origin to A.
Displacement: AB = OB − OA describes the journey from A to B.
Magnitude: For components (u, v), the length is √(u² + v²). Length has no direction.
Scalar multiple: Multiplying by a negative scalar reverses direction; the absolute value scales the length.

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).

Common 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.

Related G3 Math guides

Check vector lengths with coordinate distances

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