Loading…
Loading…
Complete IGCSE Edexcel guide to column vector notation, vector addition, subtraction, scalar multiplication, magnitude, and identifying parallel vectors. Worked examples for grades 5-7.
A column vector represents a quantity with both magnitude and direction using two stacked numbers. The top number is the horizontal component (positive = right) and the bottom number is the vertical component (positive = up). You can add, subtract, and multiply vectors by scalars by performing the operation on each component separately. The magnitude of a vector is found using Pythagoras' theorem on its components.
Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1
A vector is a quantity that has both size (magnitude) and direction. In IGCSE Maths, vectors are written using column notation with two components stacked vertically.
The top component represents horizontal displacement: positive values mean movement to the right and negative values mean movement to the left. The bottom component represents vertical displacement: positive values mean movement upward and negative values mean movement downward.
Vectors are typically labelled with bold lowercase letters (such as a or b) in printed text, or with an underline when handwritten. The vector from point A to point B is written as AB with an arrow above it.
To add two vectors, add the corresponding components separately. If a = (3, 2) and b = (1, -4), then a + b = (3 + 1, 2 + (-4)) = (4, -2).
To subtract vectors, subtract the corresponding components. Using the same vectors, a - b = (3 - 1, 2 - (-4)) = (2, 6).
Geometrically, adding two vectors means placing them end to end (the triangle rule or parallelogram rule). The resultant vector goes from the start of the first to the end of the second. Subtracting b is the same as adding -b (reversing the direction of b).
To multiply a vector by a scalar (a number), multiply each component by that number. If a = (3, -2), then 4a = (12, -8).
A positive scalar stretches the vector but keeps the same direction. A negative scalar reverses the direction and stretches. Multiplying by -1 gives the negative vector, which points in the exact opposite direction with the same magnitude.
The magnitude of a vector is its length. For a column vector (a, b), the magnitude is calculated using Pythagoras' theorem: the square root of (a squared + b squared).
For example, the vector (5, -12) has magnitude = the square root of (25 + 144) = the square root of 169 = 13. The magnitude is always a positive value.
In IGCSE notation, the magnitude of vector a is written with vertical bars on either side. If the answer is not a whole number, leave it in surd form or round to 3 significant figures as instructed.
Two vectors are parallel if one is a scalar multiple of the other. This means their components are in the same ratio. For example, (2, 6) is parallel to (1, 3) because (2, 6) = 2 times (1, 3). If the scalar is positive, both vectors point in the same direction. If negative, they point in opposite directions.
Two vectors are equal if they have exactly the same column vector — the same horizontal and vertical components. Equal vectors have the same magnitude and the same direction, regardless of their position on a grid.
Given a = (4, -1) and b = (-2, 5), find: (a) a + b, (b) 3a, (c) a - 2b.
(a) a + b = (4 + (-2), -1 + 5) = (2, 4)
(b) 3a = (3 x 4, 3 x (-1)) = (12, -3)
(c) 2b = (-4, 10), so a - 2b = (4 - (-4), -1 - 10) = (8, -11)
Answers: (a) (2, 4) (b) (12, -3) (c) (8, -11)
The vector from A to B is (6, -8). Find the magnitude of AB and determine whether AB is parallel to the vector c = (3, -4).
Step 1: Magnitude of AB = the square root of (36 + 64) = the square root of 100 = 10.
Step 2: Check for parallelism: (6, -8) = 2 times (3, -4). Since the scalar multiple is 2 (consistent for both components), AB is parallel to c.
Step 3: Since the scalar is positive, AB and c point in the same direction.
Answer: Magnitude = 10, and AB is parallel to c (same direction)
Point P has position vector (2, 5) and point Q has position vector (8, -3). Find: (a) the vector PQ, (b) the midpoint M of PQ, (c) the magnitude of PQ.
(a) PQ = position vector of Q - position vector of P = (8 - 2, -3 - 5) = (6, -8).
(b) Midpoint M = average of position vectors = ((2 + 8) / 2, (5 + (-3)) / 2) = (5, 1).
(c) Magnitude of PQ = the square root of (36 + 64) = the square root of 100 = 10.
Answers: (a) PQ = (6, -8) (b) M = (5, 1) (c) |PQ| = 10
Wrong direction for subtraction: The vector AB goes from A to B, calculated as B - A. Many students calculate A - B instead, which gives the vector BA (opposite direction).
Mixing up position vectors and displacement vectors: A position vector gives the location of a point from the origin. A displacement vector gives the journey between two points. The vector PQ equals the position vector of Q minus the position vector of P.
Forgetting to square root for magnitude: Some students calculate a squared + b squared and forget the final square root step. Always check that your magnitude answer makes geometric sense.
Sign errors in scalar multiplication: When multiplying a vector by a negative scalar, both components change sign. Forgetting to negate one component is a frequent arithmetic slip.
Draw diagrams: If a vector question does not provide a diagram, sketch one. Place vectors on a rough grid to check that your arithmetic gives sensible directions and magnitudes.
Check parallelism with ratios: To test if (a, b) is parallel to (c, d), check whether a/c = b/d (assuming c and d are non-zero). This is faster than trial-and-error scalar testing.
Leave surds in exact form: If the magnitude is not a whole number, express it as a surd (e.g. the square root of 13) unless the question asks for a decimal approximation.
Use consistent notation: Write column vectors clearly with components stacked vertically. On exam paper, ensure your handwritten vectors have clearly separated top and bottom numbers.
Vectors & Transformations Hub
All topics in this area
Vector Geometry — Proofs & Collinearity
Next topic (Higher tier)
Worksheets & Answers
Free practice for vector topics
IGCSE Tutors in Dubai
Specialist in-home IGCSE support
Maths Tutors in Dubai
Expert maths tutoring at home
Contact GetYourTutors — IGCSE Maths Vectors
Phone: (+971) 4-313-2715 | Mobile: 050-947-9432
WhatsApp: 050-947-9432
Email: info@getyourtutors.com
Emirates Towers, Office Tower, Level 41, Sheikh Zayed Road, PO Box 31003, Dubai, UAE
Last updated: March 2026
Everything you need to know about our private tutoring services in Dubai.
A dedicated tutor builds confident vector skills from column notation through to magnitude and parallel identification. In-home across Dubai.