A vector has both a size and a direction, and on a grid it is written as a column of two numbers. Transformations, such as translation, reflection and rotation, move a shape, and a vector describes the translation exactly.
This module uses the coordinates you built in coordinate geometry and the graph moves in graphs and transformations. Check the current 0580 syllabus on the Cambridge page for exactly which transformations and vector skills your paper covers, and whether they belong to Core or Extended.
What should I know before starting?
You need to read and plot coordinates in all four quadrants. You also need to add and subtract negative numbers without hesitation, and to work with simple fractions of a whole number.
If signed arithmetic is shaky, most vector errors start there. Practise it first, then come back.
An orienting example
A(1, 1). The vector a = (2, 3) takes A to B, and the vector b = (−1, 4) takes B to C. Find the coordinates of C and the single vector AC.
Step 1, find B: (1 + 2, 1 + 3) = (3, 4).
Step 2, find C: (3 + (−1), 4 + 4) = (2, 8).
Step 3, combine: AC = a + b = (2 + (−1), 3 + 4) = (1, 7).
Check: A + AC = (1 + 1, 1 + 7) = (2, 8) = C.
Three ideas appear in one short question: moving a point by a vector, chaining vectors, and checking by adding back. The lessons below take each idea on its own.
In what order should I study the lessons?
- Describe a translation using a column vector: the basic notation, with signs and direction.
- Add displacement vectors on a diagram: chaining arrows end to start to find a missing side.
- Find a point using a vector ratio: turning a ratio on a line into a fraction of a vector.
- Describe a reflection or rotation completely: knowing exactly which details each description needs.
- Distinguish position from displacement: the difference between a vector from the origin and one between two points.
Then try the vectors and transformations practice set, which mixes all five skills. The non-calculator working trainer supports the exact arithmetic.
What are the common traps?
- Reversing a vector. AB and BA have opposite signs. Say “end minus start”.
- Using ratio numbers as the fraction. A ratio of 1:2 means one third of the line, not one half.
- Incomplete descriptions. A rotation needs an angle, a direction and a centre. A reflection needs the line.
- Mixing points and vectors. Write points with a capital letter, such as P(3, −1), and vectors as columns.
How do I use the practice set?
Start with the first four questions to check the notation, then move to the ratio and transformation questions. Write every step before you open the answer.
Note each wrong answer in the mistake log and retest queue, and revisit the corresponding lesson before trying again a few days later.
If vectors keep slipping despite practice, our teachers can look at your working in online one-to-one Mathematics tuition.