Vectors and transformation geometry for Edexcel IGCSE Mathematics Specification B: Methods and Worked Examples

Vectors and transformation geometry bring together algebra and geometry in a way that many students find surprisingly satisfying once the notation clicks. If you are preparing for the Edexcel IGCSE Mathematics Specification B exam, this section is well worth the investment.

The edexcel igcse mathematics specification b vectors and transformation geometry content is split into two closely related areas: Vectors and Transformations. Vectors give you a precise algebraic language for describing movement and position. Transformations give you a systematic way to describe how shapes change under reflections, rotations, translations, and enlargements. Both topics connect to the Matrices section of the specification, and together they form a powerful toolkit.

Vectors

Scalar and vector quantities

A scalar has magnitude only (for example, speed, temperature, mass). A vector has both magnitude and direction (for example, velocity, displacement, force). The distinction matters: 5 m/s is a speed; 5 m/s due north is a velocity.

Notation

Vectors can be written as column vectors, bold letters, or with an arrow above. In this article and in the vectors and transformation geometry edexcel igcse exam, column vector notation is the most common:

a = (3 over -2) means 3 units to the right and 2 units down.

A directed line segment from A to B is written as the vector AB, and represents the journey from A to B.

Adding and subtracting vectors

Vectors are added component by component:

(3 over 1) + (2 over -4) = (5 over -3).

Subtracting reverses the direction of the second vector:

(3 over 1) - (2 over -4) = (1 over 5).

Scalar multiplication

Multiplying a vector by a scalar scales its magnitude without changing its direction (unless the scalar is negative, which reverses the direction):

3 x (2 over -1) = (6 over -3).

The modulus (magnitude) of a vector

The modulus of a = (x over y) is |a| = sqrt(x² + y²).

Worked example: Find the modulus of the vector (5 over -12).

|a| = sqrt(25 + 144) = sqrt(169) = 13.

Parallel vectors

Two vectors are parallel if one is a scalar multiple of the other. If b = ka for some scalar k, then a and b are parallel. If k > 0, they point in the same direction; if k < 0, they point in opposite directions.

Worked example: Show that p = (4 over 6) and q = (10 over 15) are parallel.

q = 2.5 x p, since 10 = 2.5 x 4 and 15 = 2.5 x 6. Since q is a scalar multiple of p, they are parallel.

Position vectors

The position vector of a point P is the vector from the origin O to P, written as the vector OP. If A has position vector a and B has position vector b, then the vector AB = b - a.

Worked example: A has position vector (1 over 3) and B has position vector (5 over 7). Find the vector AB and its modulus.

AB = b - a = (5-1 over 7-3) = (4 over 4).

|AB| = sqrt(16 + 16) = sqrt(32) = 4sqrt(2).

Resultant vectors

The resultant of two or more vectors is their sum. When forces or displacements act together, the resultant gives the overall effect.

Worked example: Points P, Q, R have position vectors p = (1 over 2), q = (4 over 5), r = (10 over 11). Show that P, Q, and R are collinear.

PQ = q - p = (3 over 3). QR = r - q = (6 over 6) = 2 x (3 over 3) = 2 x PQ.

Since QR = 2PQ, the vectors are parallel and share point Q, so P, Q, and R are collinear.

Transformations

The four transformations

The specification covers four types of transformation, and you need to be able to perform them and describe them fully.

Missing any required piece of information loses marks, even if the diagram is correct.

Reflections

Worked example: Reflect the point (3, 2) in the line y = x.

Under reflection in y = x, the coordinates swap: (3, 2) becomes (2, 3).

For reflection in the x-axis, negate the y-coordinate: (3, 2) becomes (3, -2).

For reflection in the y-axis, negate the x-coordinate: (3, 2) becomes (-3, 2).

Rotations

Worked example: Rotate the point (4, 1) by 90 degrees anticlockwise about the origin.

Using the rule (x, y) maps to (-y, x): (4, 1) becomes (-1, 4).

For 180 degrees about the origin: (x, y) maps to (-x, -y). (4, 1) becomes (-4, -1).

Translations

A translation shifts every point by the same column vector. The edexcel igcse mathematics specification b revision notes for translations should emphasise that shape, size, and orientation are all preserved; only position changes.

Worked example: Translate triangle with vertices (1, 2), (3, 2), (2, 5) by the vector (4 over -3).

New vertices: (5, -1), (7, -1), (6, 2).

Enlargements

An enlargement changes the size of a shape by a scale factor from a centre of enlargement. If the scale factor is between 0 and 1, the image is smaller. If the scale factor is negative, the image is on the opposite side of the centre.

Worked example: Enlarge triangle A(1, 1), B(3, 1), C(1, 4) by scale factor 2, centre the origin.

Multiply each coordinate by 2: A'(2, 2), B'(6, 2), C'(2, 8).

Worked example: Enlarge point P(5, 3) by scale factor -1 from centre (2, 1).

Vector from centre to P: (3 over 2). Multiply by -1: (-3 over -2). Image = (2, 1) + (-3, -2) = (-1, -1).

Combining transformations

The edexcel igcse mathematics specification b notes on combined transformations require you to find the single transformation equivalent to two successive transformations. This links directly to the Matrices topic.

Worked example: A shape is reflected in the y-axis, then reflected in the x-axis. What single transformation is equivalent?

Under the first reflection, (x, y) becomes (-x, y). Under the second, (-x, y) becomes (-x, -y). The combined effect is (x, y) maps to (-x, -y), which is a rotation of 180 degrees about the origin.

Matrix transformations

The specification also includes multiplying a position vector by a 2 x 2 matrix to produce a transformation. This is covered in depth in the Matrices article, but the connection is direct: each column of the matrix is the image of a unit vector, and multiplying the matrix by a position vector gives the transformed coordinates.

Common mistakes in Vectors and Transformations

• Direction of AB versus BA. The vector AB goes from A to B. The vector BA = -AB, going from B to A. Mixing these up reverses the sign of every component.
• Forgetting to state the direction of rotation. "90 degrees" is incomplete. You must say "90 degrees anticlockwise" or "90 degrees clockwise."
• Scale factor errors in enlargement. A scale factor of 2 doubles distances from the centre; a scale factor of 1/2 halves them. A negative scale factor produces an image on the opposite side of the centre.
• Treating translation as multiplication. A translation adds a vector to every point. It does not multiply coordinates. This is different from an enlargement about the origin, which does multiply.
• Collinearity proofs without conclusion. Showing that two vectors are parallel is necessary but not sufficient for collinearity. You must also show that the vectors share a common point (which they do if one starts where the other ends).

Self-check questions

Try these edexcel igcse mathematics specification b practice questions before looking at the answers.

1. Given a = (2 over -3) and b = (-1 over 4), find 3a - 2b and its modulus.
2. Points A(2, 5) and B(8, 13) are given. M is the midpoint of AB. Find the position vector of M.
3. Describe fully the single transformation that maps triangle P(1, 1), Q(3, 1), R(1, 3) to P'(-1, 1), Q'(-3, 1), R'(-1, 3).
4. Show that the points with position vectors (2 over 1), (5 over 7), and (7 over 11) are collinear.

Solutions

1. 3a = (6 over -9). 2b = (-2 over 8). 3a - 2b = (8 over -17). Modulus = sqrt(64 + 289) = sqrt(353) = 18.79 (to 4 s.f.).

2. M = ((2+8)/2 over (5+13)/2) = (5 over 9).

3. Each x-coordinate has been negated, y-coordinates unchanged. This is a reflection in the y-axis.

4. Let A = (2 over 1), B = (5 over 7), C = (7 over 11). AB = (3 over 6). BC = (2 over 4). BC = (2/3) x AB. Since BC is a scalar multiple of AB and they share point B, A, B, and C are collinear.

Vectors and transformations reward careful, methodical work. Every calculation should be written out clearly, every transformation described in full, and every proof structured logically. The edexcel igcse mathematics specification b explained techniques in this section are ones you can rely on: once you understand the notation and the rules, the answers follow naturally from the method.