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CIE A-Level Maths Study Notes

2.7.4 Equation of a Line in Vector Terms

In this comprehensive exploration of the equation of a line in vector terms, we delve into the intricacies of vector algebra and its application in defining lines in both two and three-dimensional spaces. This knowledge is fundamental for students in understanding complex geometrical relationships and solving advanced mathematical problems.

Vector Equation of a Line

The vector equation of a line in space is expressed as r=a+tb\mathbf{r} = \mathbf{a} + t\mathbf{b}. This equation is pivotal in vector algebra, representing a line in terms of vectors. Here:

vector equation of a line

Image courtesy of Cuemath

Finding a Line's Equation:

1. Position Vectors: Start with position vectors of two points, a\mathbf{a} and b\mathbf{b}.

2. Direction Vector: Use the difference ba\mathbf{b} - \mathbf{a} as the direction vector.

3. Line Equation: Write the equation as r=a+t(ba)\mathbf{r} = \mathbf{a} + t(\mathbf{b} - \mathbf{a}), with tt varying to trace the line.


Example 1: Finding a Point on a Line

Given the line r=3i+4j+t(2ij+3k)\mathbf{r} = 3\mathbf{i} + 4\mathbf{j} + t(2\mathbf{i} - \mathbf{j} + 3\mathbf{k}), find the position vector of a point on the line when t=2t = 2.


1. Substitute t=2t = 2 into the equation.

2. Calculate the position vector: r=3i+4j+2(2ij+3k)=7i+2j+6k\mathbf{r} = 3\mathbf{i} + 4\mathbf{j} + 2(2\mathbf{i} - \mathbf{j} + 3\mathbf{k}) = 7\mathbf{i} + 2\mathbf{j} + 6\mathbf{k}.

3. The position vector of the point is 7i+2j+6k 7\mathbf{i} + 2\mathbf{j} + 6\mathbf{k}.

Example 2: Equation of a Line Through Two Points

Find the equation of the line passing through the points with position vectors 2i+3j2\mathbf{i} + 3\mathbf{j} and i+4j+5k.-\mathbf{i} + 4\mathbf{j} + 5\mathbf{k}.


1. Let a=2i+3j\mathbf{a} = 2\mathbf{i} + 3\mathbf{j} and b=i+4j+5k\mathbf{b} = -\mathbf{i} + 4\mathbf{j} + 5\mathbf{k}.

2. The equation of the line is r=a+t(ba)=2i+3j+t(3i+j+5k)\mathbf{r} = \mathbf{a} + t(\mathbf{b} - \mathbf{a}) = 2\mathbf{i} + 3\mathbf{j} + t(-3\mathbf{i} + \mathbf{j} + 5\mathbf{k}).

3. This represents all points on the line as a linear combination of a\mathbf{a} and b\mathbf{b}.

Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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