Sheet № 56 · Higher only · AQA · Edexcel · OCR
Vectors –
Vectors is a Higher tier GCSE Maths topic that appears on AQA, Edexcel, and OCR papers. Unlike coordinates, which describe a fixed position, a vector describes a movement — it has both magnitude (size) and direction. You need to be comfortable with column vector notation, adding and subtracting vectors, multiplying by a scalar, and using
§Key definitions
Question:
OA = a and OB = b. M is the midpoint of AB. Find the vector OM in terms of a and b.
Answer:
OM = ½(a + b)
Key point:
The method is to express both vectors from O and check for a scalar multiple. If OP = kOM, they are collinear.
Question 1:
Write the column vector for a translation of 4 units left and 3 units up.
Question 2:
Given a = (2 over 5) and b = (−1 over 3), find a + b and 2a − b.
§Formulas to memorise
Vector addition: (a over b) + (c over d) = (a+c over b+d)
Vector subtraction: (a over b) − (c over d) = (a−c over b−d)
Scalar multiplication: k × (a over b) = (ka over kb)
Magnitude of (x over y) = √(x² + y²)
Column vector: — Written as a vertical pair of numbers in brackets, e.g. (3 over 2) means "3 units right and 2 units up."
Letter notation: — A single bold or underlined letter such as a or a with a squiggle underneath. In handwriting, draw a squiggle below the letter.
Equal vectors — have the same magnitude and direction. They do not need to start at the same point.
Parallel vectors — point in the same (or exactly opposite) direction. One is a scalar multiple of the other. If b = ka for some scalar k, then a and b are parallel.
Collinear points — lie on the same straight line. To prove three points are collinear, show that the vector from one to another is a scalar multiple of the vector from one to a third.
Worked example
OA = a and OB = b. M is the midpoint of AB. Find the vector OM in terms of a and b.
Working:
⚠ Common mistakes
- ✗Forgetting to negate. When travelling from B to A, the vector is −AB, not AB. Getting the sign wrong is the most frequent error.
- ✗Not simplifying. Always collect like terms and write your final vector in its simplest form.
- ✗Confusing position vectors with direction vectors. OA is a position vector (from the origin to A). AB is a direction vector (from A to B). They are different.
- ✗Stating collinear without a common point. Parallel vectors alone do not prove collinearity — you must also show the points share a common point on the line.
- ✗Mixing up scalar multiples. If b = 3a, the vectors are parallel and b is three times the length of a. But b = a + 3 is not a valid vector equation — you cannot add a scalar to a vector.
✦ Exam tips
- →Draw the diagram and label every vector. This helps you plan your route.
- →Use the "go via" approach. To get from X to Y, travel via known points: XY = XA + AY.
- →Show all steps in proof questions. Marks are awarded for method — setting up the route, simplifying, and concluding.
- →State your conclusion clearly. "Since OP = 2OQ, O, P, and Q are collinear" earns the final mark.
- →Practise midpoint results. The vector to the midpoint of AB is always ½(a + b) when O is the origin — this shortcut saves time.