Vectors — AQA GCSE Mathematics

Vectors are quantities with both magnitude and direction, represented as column vectors or directed line segments. At AQA GCSE Mathematics, you must be able to add, subtract and multiply vectors by scalars, prove geometric relationships such as parallel lines and collinearity, and solve problems using vector notation.

Vectors are fundamental in mathematics and physics, representing quantities that have both magnitude and direction. In AQA GCSE Mathematics, you will learn to represent vectors as column vectors, perform addition and subtraction, multiply by scalars, and use vectors to solve geometric problems involving parallel lines and collinearity.

This topic extends your understanding of geometry and algebra, providing tools to prove relationships in shapes such as parallelograms and triangles. It also forms a foundation for further study in mechanics and vectors at A-level. Mastery of vectors enhances problem-solving skills and spatial reasoning.

• →A vector has both magnitude and direction, represented as a column vector (x, y) or as a directed line segment with an arrow.
• →Vector addition: a + b is found by placing the tail of b at the head of a (triangle law) or by adding corresponding components.
• →Scalar multiplication: multiplying a vector by a scalar changes its magnitude but not its direction (unless scalar is negative, which reverses direction).
• →Parallel vectors are scalar multiples of each other: if a = k b, then a and b are parallel.
• →Collinearity: points A, B and C are collinear if vector AB is a scalar multiple of vector BC and they share a common point.

• the word translation, which a full description needs alongside the vector
• a correct column vector, with the across value on top, the up value underneath and both entries carrying the right sign
• moving every vertex of the object by the vector when you are asked to carry a translation out, even if one point is then plotted wrongly
• a correctly drawn image, the same size and the same way round as the object
• a correct column vector after adding or subtracting, with both entries right
• a correct scalar multiple, with every entry multiplied rather than only the top one
• a correct route across a diagram, such as writing the journey from A to B then B to C, even if the final vector is wrong
• collecting like terms in the answer, for example gathering 2a and 3a into 5a
• a correct route written in terms of the two named vectors, even before it is tidied up
• correct use of a midpoint or a ratio to split a vector, such as taking ⅖ of a side
• simplifying far enough to show one vector as a multiple of another
• the concluding statement, naming parallel lines or points on a straight line and giving the reason

• 💡Count squares from one corner of the object to the very same corner of the image, never to a different corner.
• 💡Check each sign against the axes: right and up are positive, left and down are negative.
• 💡Name one transformation only when you describe how a shape has moved.
• 💡Draw the route on the diagram with arrows before you write any algebra, so the direction of every step is clear.
• 💡Travelling against an arrow means using the negative of that vector.
• 💡Underline your vector letters so you never confuse a vector with a plain number.
• 💡Mark the arrows on the diagram and write each route as a chain of letters before you turn it into a and b.
• 💡When two results turn out to be multiples of each other, say so in words and name the point the two lines share.
• 💡Keep fractions exact and gather the a terms and the b terms separately.
• 💡Always show your working clearly, especially when using vector notation. Write vectors with arrows or underlines to avoid confusion with lengths.
• 💡When proving geometric statements, use precise language: state that vectors are parallel, mention the scalar multiple, and identify shared points for collinearity.
• 💡Check your answers by considering if the resultant vector makes sense in terms of direction and magnitude. For example, if adding two vectors, the result should be a diagonal of the parallelogram formed.

• writing the up value on top and the across value underneath, so the column vector is the wrong way round
• describing the movement only in words, such as five right and two down, when a column vector is asked for
• naming two transformations, for example translation and reflection, which loses the description mark
• counting between vertices that do not match, so the vector points the right way but is the wrong size
• subtracting the wrong way round, working out b − a when the route calls for a − b
• multiplying only the top entry of a column vector by the scalar and leaving the bottom one alone
• joining vectors head to head rather than nose to tail, which quietly reverses one of them
• treating a vector as a length, so ignoring the direction when two vectors have the same size
• reading AP:PB = 2:3 as AP being ⅔ of AB, when the line has been cut into five equal parts; correct by taking AP as ⅖ of AB
• travelling in the direction of an arrow when the route needs the reverse, which loses a minus sign; correct by checking each step against the arrows
• stopping once the algebra is done and never stating the geometric conclusion, so the proof is unfinished; correct by naming the parallel lines or collinear points and giving the reason
• Students often think that vectors can be added by simply adding their magnitudes, ignoring direction. Correction: Vectors must be added using the triangle law or by adding components.
• Students may believe that if two vectors are parallel, they must be equal. Correction: Parallel vectors are scalar multiples, not necessarily equal; they can have different magnitudes or opposite directions.
• Students sometimes forget that vector AB = - vector BA. Correction: Reversing the direction of a vector changes its sign.

1. 1Week 1: Start by reviewing the basics of vectors: notation, column vectors, magnitude, and direction. Practice adding and subtracting vectors using both graphical and component methods.
2. 2Week 1: Learn about scalar multiplication and parallel vectors. Solve problems involving finding missing vectors in shapes like parallelograms and triangles.
3. 3Week 2: Focus on geometric proofs using vectors, especially collinearity and parallel lines. Work through past paper questions on these topics.
4. 4Week 2: Complete a mixed set of exam-style questions under timed conditions. Review mistakes and revisit any weak areas.
5. 5Ongoing: Use flashcards for key definitions and formulas, and practice drawing diagrams to visualise vector problems.

• 📋Column vector arithmetic: Questions asking to add, subtract, or multiply vectors by scalars, often in component form. Advice: Be careful with signs and order of operations.
• 📋Geometric proofs: Prove that points are collinear, lines are parallel, or find ratios of lengths using vectors. Advice: Always express vectors in terms of given variables and look for scalar multiples.
• 📋Vector geometry in shapes: Find vectors for diagonals or midpoints in parallelograms, triangles, or trapezia. Advice: Use the triangle law and properties of the shape.
• 📋Problem-solving with vectors: Multi-step problems combining vector arithmetic and geometric reasoning. Advice: Draw a clear diagram and label all known vectors.