Pearson Edexcel · GCSE · Mathematics
Course version: 1MA1 · Version from Mar 2026
Apply addition and subtraction of vectors, multiplication of vectors by a…
Full specification wording
Geometry and measures - Vectors: apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors; use vectors to construct geometric arguments and proofs (Higher tier)
Vectors have magnitude and direction, represented diagrammatically (e.g. a, b) or as column vectors.
Checked against the Pearson Edexcel 1MA1 specification (opens in a new tab) on .
- Point 3 of 3
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
Vectors have magnitude and direction, represented diagrammatically (e.g. a, b) or as column vectors. Arithmetic applies to both. For diagrams, find a route between points by adding or subtracting vectors; travelling against a vector's direction negates it (e.g. BA = -AB). For column vectors, add or subtract corresponding components, and multiply each component by a scalar, e.g. 3 * (2, -1) = (6, -3). Geometric proofs use these skills. To prove lines AB and CD are parallel, show that vector AB is a scalar multiple of vector CD (i.e. AB = kCD). To prove points X, Y and Z are collinear (on the same straight line), show that one vector between them is a scalar multiple of another and that they share a common point, e.g. show XY = kXZ.
What earns the marks
- A correct route for a vector written in terms of the given vectors, e.g., AB = AO + OB.
- Correctly simplifying a vector expression by collecting like terms.
- Correctly performing column vector arithmetic (addition, subtraction, or scalar multiplication).
- A concluding statement that correctly interprets the vector relationship, e.g., '...so the vectors are parallel as one is a scalar multiple of the other'.
Common mistakes and how to put them right
Forgetting to negate a vector when travelling against its defined direction on a diagram.
Misinterpreting a ratio; e.g., if a point divides a line in the ratio 2:3, the fractions of the whole line are 2/5 and 3/5, not 2/3.
Stopping after finding a scalar multiple relationship (e.g., AB = 2CD) without writing a concluding sentence to explain what has been proved (parallelism or collinearity).
Exam tips
- On diagrams, draw arrows on the lines to show the route you are taking. This helps avoid sign errors when travelling against a vector's direction.
- A proof question requires a final concluding sentence. State clearly what you have shown (e.g., 'The lines are parallel') and why ('because one vector is a multiple of the other').
Try the lesson
The start of the Geometry and measures - Vectors: apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors; use vectors to construct geometric arguments and proofs (Higher tier) lesson, made from checked teaching notes for this specification point.
Have a go first
The vector a is given by a = (4, -1) and the vector b is given by b = (-2, 3). Work out the column vector 2a - 3b.
Listen
Vectors have magnitude and direction, represented diagrammatically (e.g. a, b) or as column vectors. Arithmetic applies to both. For diagrams, find a route between points by adding or subtracting vectors; travelling against a vector's direction negates it (e.g. BA = -AB). For column vectors, add or subtract corresponding components, and multiply each component by a scalar, e.g. 3 * (2, -1) = (6, -3). Geometric proofs use these skills. To prove lines AB and CD are parallel, show that vector AB is a scalar multiple of vector CD (i.e. AB = kCD). To prove points X, Y and Z are collinear (on the same straight line), show that one vector between them is a scalar multiple of another and that they share a common point, e.g. show XY = kXZ. A strong answer earns marks for these points. A correct route for a vector written in terms of the given vectors, e.g., AB = AO + OB. Correctly simplifying a vector expression by collecting like terms. Correctly performing column vector arithmetic (addition, subtraction, or scalar multiplication). A concluding statement that correctly interprets the vector relationship, e.g., '...so the vectors are parallel as one is a scalar multiple of the other'. Exam tip: On diagrams, draw arrows on the lines to show the route you are taking. This helps avoid sign errors when travelling against a vector's direction.
Key words
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Vector arithmetic (diagrammatic and column)
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Geometric proofs
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Parallel and collinear vectors
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Ratios on a line
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See every definition
- Vector arithmetic (diagrammatic and column)
- Addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors.
- Geometric proofs
- Arguments and proofs constructed using vectors, such as showing one vector is a scalar multiple of another.
- Parallel and collinear vectors
- Lines are parallel when one vector is a scalar multiple of the other; points are collinear when one vector between them is a scalar multiple of another and they share a common point.
- Ratios on a line
- A point dividing a line in a given ratio, where the fractions of the whole line are taken from the total of the ratio, not one part over another.
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