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Vectors

FoundationHigherAQAEdexcelOCR

This free Foundation and Higher GCSE Maths worksheet on vectors helps you revise vector arithmetic and geometric vector proofs. Questions build from straightforward to exam standard, with full worked answers in the mark scheme — ideal for revision or homework. Suitable for AQA, Edexcel and OCR. Vectors with the same length and direction are equal, wherever they sit.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Vectors have both magnitude and direction, which distinguishes them from ordinary numbers. In geometry problems they are usually written as bold letters or with an arrow above them.

The vector from \(A\) to \(B\) is written \(\overrightarrow{AB}\). Reversing the direction reverses the sign, so \(\overrightarrow{BA} = -\overrightarrow{AB}\).

Journeys combine by addition. Going from \(A\) to \(B\) then \(B\) to \(C\) is the same as going directly from \(A\) to \(C\), so \(\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}\). Most vector geometry questions are built on finding a route between two points using the vectors you are given.

Revision notes

Notation and direction

\(\overrightarrow{AB}\) is the vector from \(A\) to \(B\), and \(\overrightarrow{BA} = -\overrightarrow{AB}\).

Reversing the journey reverses the sign, which is often needed to build a route from the vectors given.

Finding a route

Travel from the start to the end using any path made of known vectors, adding them as you go.

If \(\overrightarrow{AB} = \mathbf{a}\) and \(\overrightarrow{BC} = \mathbf{b}\), then \(\overrightarrow{AC} = \mathbf{a} + \mathbf{b}\). Going backwards along a vector subtracts it.

Parallel vectors

Two vectors are parallel if one is a multiple of the other.

So \(2\mathbf{a}\) is parallel to \(\mathbf{a}\) and twice as long. Showing that one expression is a multiple of another proves the lines are parallel, which is a common exam requirement.

Key points

  • Vectors have magnitude and direction.
  • \(\overrightarrow{AB}\) goes from A to B.
  • \(\overrightarrow{BA} = -\overrightarrow{AB}\).
  • \(\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}\).
  • Going backwards along a vector subtracts it.
  • Parallel vectors are multiples of each other.

Worked examples

Example 1

If \(\overrightarrow{AB} = \mathbf{a}\) and \(\overrightarrow{BC} = \mathbf{b}\), find \(\overrightarrow{AC}\).

Working

\[A \to B \to C\]find a route using known vectors
\[\mathbf{a} + \mathbf{b}\]add the vectors along the route

Example 2

If \(\overrightarrow{AB} = \mathbf{a}\), write \(\overrightarrow{BA}\).

Working

\[\text{Reversing the journey reverses the sign}\]apply the rule
\[-\mathbf{a}\]state the vector

Example 3

Show that \(4\mathbf{a}\) is parallel to \(\mathbf{a}\).

Working

\[4\mathbf{a} = 4 \times \mathbf{a}\]one is a multiple of the other
\[\text{So they are parallel}\]state the conclusion

Common mistakes

  • Forgetting to reverse the sign when travelling backwards.

    Going from B to A gives −AB, not AB.

  • Adding vectors that do not form a continuous route.

    The end of one must be the start of the next.

  • Treating vectors as ordinary numbers.

    Direction matters, so the order and signs cannot be ignored.

  • Not stating the conclusion in a parallel proof.

    Show the multiple, then say explicitly that the vectors are parallel.

Exam tips

  • Sketch the route on the diagram before writing anything.
  • Reverse the sign whenever you travel against a given vector.
  • Simplify the expression fully at the end.
  • For parallel proofs, show one is a multiple of the other and say so.

Key terms

Vector
A quantity with magnitude and direction.
Magnitude
The length of a vector.
Resultant
The single vector equivalent to several combined.
Parallel
Vectors that are multiples of each other.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.