GCSE Maths: Vectors

Vectors start simply at GCSE, with translations written as column vectors, and end with some of the most demanding reasoning on a Higher paper, where you prove that points lie on a straight line or that two lines are parallel. This material follows the vectors content of the GCSE subject content for England: describing translations as 2D vectors, adding and subtracting vectors, multiplying by a scalar, and, for Higher, using vectors to build geometric arguments and proofs.

The quiz has twelve questions. The first group is column-vector arithmetic: adding vectors, working out a combination such as $2\mathbf{a} - 3\mathbf{b}$, writing the translation from one point to another, finding a vector between two position vectors and its magnitude with Pythagoras' theorem, and spotting which vectors are parallel to a given one. The second group is Higher-tier vector geometry with every set-up described in words: the vector $\overrightarrow{AB}$ in a triangle $OAB$, the position vector of a midpoint, the position vector of a point that divides a line in a given ratio, finding an unknown so that two vectors are parallel, and deciding which statement really proves that three points are collinear. The wrong options are the usual traps: subtracting the wrong way round, splitting a ratio the wrong way, and accepting a statement that is true for any three points as a proof.

The flashcards cover the vocabulary and the rules you need to write clear proofs: scalar, magnitude, position vector, resultant, parallel and collinear, the rules for adding, subtracting and scaling column vectors, the vector $\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$, and the midpoint result.

The printable written work has eight problems to solve with full working: combining column vectors and finding a magnitude, a vector between two points with its length and midpoint, a vector with a given direction and length, finding unknown components of parallel vectors, a point dividing a line in the ratio 3 : 1, a proof that the diagonals of a parallelogram bisect each other, a collinearity proof, and a proof that the line joining two midpoints in a quadrilateral is parallel to a pair of sides. These proofs reward a clear chain of steps: name the route you take, simplify, factorise to show a multiple, then state the conclusion in words.

Foundation students should concentrate on the column-vector questions; the geometry and proof items are Higher-tier content. The material is aimed at Year 10 and Year 11.

• Add, subtract and multiply column vectors by a scalar
• Describe a translation as a column vector and find the vector between two points
• Find the magnitude of a vector using Pythagoras' theorem
• Recognise parallel vectors as scalar multiples and find unknown components
• Express vectors in a shape in terms of given vectors, including midpoints and ratios (Higher)
• Prove that points are collinear or that lines are parallel using vectors (Higher)

Practice material written by Zestly, based on the DfE GCSE mathematics subject content (DFE-00233-2013), Geometry and measures: vectors (G24, G25).