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The vector equation of a line, r = a + λb, describes every point on a line using a position vector and a direction vector. This topic covers writing lines in vector form and testing whether a point lies on a line. It's a key topic in A-Level and International A-Level (iAL) Pure Mathematics. This free interactive course teaches the method with worked examples and gives instant-feedback practice questions to lock it in.
How you do it
The equation is a position vector plus a scalar times a direction vector. Any point on the line works as the position; subtracting two points gives the direction. Different-looking equations can describe the same line.
Before you start
The prerequisite is vectors and coordinates. Examined in Pure 4. The same ideas return in intersections, skew lines and angles. The topic sits within Vectors.
What you learn
A straight line is determined by one point on it and its direction. The same method finds an unknown coordinate of a point that is known to lie on the line. Parallel direction vectors alone do not prove collinearity — you must also state the common point, otherwise the two segments could be on parallel lines. The course works through the magnitude of a vector, unit vectors, position vectors and the vector between two points, the vector equation of a line, testing whether a point lies on a line and problem solving: points on lines and collinearity. Every section has instant-feedback practice questions matched to Edexcel International A-Level and 9MA0 Pure, and the course finishes with full exam-style questions you mark yourself.
Related Topics
In our A-level maths courses, Vector Equations of Lines belongs to Vectors and is met in the second year of A-level. You work through 6 concepts and 6 worked examples, testing yourself at 6 points. Nearby topics in the same strand are Angles & Geometric Problems , Parallel, Intersecting & Skew Lines and Scalar (Dot) Product. The Vectors revision engine gives you as many extra questions as you want, tracked on your skill tree.
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