A-Level Vectors Revision
Vectors revision for A-Level and International A-Level Maths
Vectors is a short topic with a high mark density: a single Pure 4 question can carry most of the topic's content, and it usually hinges on one idea — that a line is a starting point plus a direction, and everything else follows from comparing directions. This free revision engine covers the vectors content of the International A-Level Pure 4 paper and the equivalent Edexcel A-Level material, in both two and three dimensions.
What the vectors revision engine covers
Vector equations of lines — writing a line as a position vector plus a multiple of a direction vector, finding points on a line, and testing whether a given point lies on it Parallel, intersecting and skew lines — deciding which of the three a pair of lines is, and finding the point of intersection where one exists The scalar product — calculating it, using it to test for perpendicularity, and finding an unknown component Angles and geometric problems — the angle between two vectors or two lines, angles at a vertex of a triangle, and magnitudes
Vector formulas and results you need to know
Equation of a line. r = a + t d, where a is the position vector of a point on the line and d is its direction. Direction between two points. The direction from A to B is the position vector of B minus that of A. Magnitude. The magnitude of a vector is the square root of the sum of the squares of its components. Unit vector. Divide a vector by its own magnitude. The scalar product. Multiply matching components and add: a₁b₁ + a₂b₂ + a₃b₃. Perpendicular vectors. Two vectors are perpendicular exactly when their scalar product is zero. Angle between vectors. cos θ = (a · b) divided by the product of the two magnitudes. For the acute angle between two lines, take the modulus of the scalar product first. Parallel lines. Two lines are parallel when one direction vector is a scalar multiple of the other. If they are parallel but no point is shared, they are distinct; if not parallel and they never meet, they are skew.
How to use it
Pick a topic and the engine generates a question. Column vectors can be typed in whichever form you find easiest — ve(1,-2,3), plain brackets (1,-2,3), just the components, or i, j, k form — and all are marked as correct. There is an input guide in the engine showing each format. Two hints per question point you at the right method without giving the answer.
More A-Level Maths revision
Vectors shares its underlying ideas with coordinate geometry: a direction vector is a gradient in disguise, and the perpendicularity test is the same fact written differently. See every topic and track your progress on the revision hub. Everything here is free to use.