Equation of a line
\[\vec r=\vec a+\lambda\vec b,\qquad \frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}\]
Three Dimensional Geometry
Equation of a Line in Space
Understand
A line through the point with position vector \(\vec a\) and parallel to \(\vec b\) has vector equation \(\vec r=\vec a+\lambda\vec b\), \(\lambda\in\mathbb{R}\). If \(\vec a=(x_1,y_1,z_1)\) and \(\vec b\) has direction ratios \(a,b,c\):
\[\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}.\]
Through two points \(\vec a\) and \(\vec b\): \(\vec r=\vec a+\lambda(\vec b-\vec a)\), i.e. \(\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}\).
Reading a Cartesian equation: write it as \(\frac{x-p}{a}=\dots\); the point is \((p,q,r)\) and the direction ratios are the denominators. Watch for forms like \(\frac{2x-1}{4}\), which must first become \(\frac{x-\frac12}{2}\).
General point: every point of the line is \((x_1+a\lambda,\ y_1+b\lambda,\ z_1+c\lambda)\) – use this to find intersections or the foot of a perpendicular.
Key Concepts
- Vector form: r = a + λb.
- Cartesian: (x − x₁)/a = (y − y₁)/b = (z − z₁)/c.
- Coefficient of x, y, z must be 1 before reading direction ratios.
Formula Bank
Key Points
General point on a line
On \(\frac{x-1}{2}=\frac{y+3}{-1}=\frac{z}{4}=\lambda\), every point is \((1+2\lambda,\ -3-\lambda,\ 4\lambda)\).
Common Mistakes
Reading DRs from non-standard forms
In \(\frac{1-x}{2}=\frac{y}{3}=\frac{2z+1}{4}\), rewrite as \(\frac{x-1}{-2}=\frac{y}{3}=\frac{z+\frac12}{2}\): DRs are \(-2,3,2\).
Practice & Topic Test
Topic Practice
A fresh set of questions from this topic. Change answers freely – solutions appear after you submit.
Start PracticeTopic Summary
Point + direction give the line; convert between vector and Cartesian forms and use the general point for calculations.
Ready to practise this topic?
Create Student Account
Create a free account to take practice sessions and tests, see detailed explanations and track your progress.