Equations of Lines in 3D (Edexcel A-Level Further Mathematics): Revision Notes

6.1.1 Equations of Lines in 3D

Introduction

In 3D space, a straight line can be described in multiple forms:

Vector Form:

$${r} = \mathbf{a} + \lambda \mathbf{b}$$

Cartesian Form:

$$\frac{x-a_1}{b_1} = \frac{y-a_2}{b_2} = \frac{z-a_3}{b_3}$$

Both forms are equivalent and represent the same line.

Vector Form of a Line

The vector equation of a line is written as:

$${r} = \mathbf{a} + \lambda \mathbf{b}$$

where:

• $r$ is the position vector of any point on the line.
• $a$ is the position vector of a fixed point on the line.
• $b$ is the direction vector of the line.
• $\lambda$ is a scalar parameter.

Cartesian Form of a Line

The Cartesian form is derived from the vector form:

$$\mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \implies x = a_1 + \lambda b_1, \; y = a_2 + \lambda b_2, \; z = a_3 + \lambda b_3$$

Eliminating $\lambda$ gives:

$$\frac{x - a_1}{b_1} = \frac{y - a_2}{b_2} = \frac{z - a_3}{b_3}$$

Worked Examples

Note Summary