AI Tutor
If a satellite fires a thruster in deep space, its path is a laser-straight trajectory through three dimensions. How can we write a single mathematical rule that pinpoints every coordinate the satellite will ever visit?
In 2D, we relied on y=mx+cy = mx + c, but a single slope number cannot describe tilting across three perpendicular axes simultaneously. What replaces slope in 3D?

The Anatomy of a 3D Line

To reach any point RR on a line, you start at the origin OO, travel along a position vector a\mathbf{a} to an anchor point on the line, and then travel some scaled amount along a direction vector d\mathbf{d}.
This geometric path gives the vector equation of a line: r=a+Ξ»d\mathbf{r} = \mathbf{a} + \lambda \mathbf{d}, where r=(xyz)\mathbf{r} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} represents the position vector of any general point on the line, and Ξ»\lambda is a scalar parameter (any real number).
In the 1880s, American physicist Josiah Willard Gibbs developed this vector formulation to streamline complex electromagnetic calculations into concise geometric steps.
How do we assemble this equation when given specific 3D coordinates and vector components?

Worked Example: Building the Equation

Find a vector equation for the straight line passing through point A(2,βˆ’1,4)A(2, -1, 4) parallel to the direction vector d=3iβˆ’2j+5k\mathbf{d} = 3\mathbf{i} - 2\mathbf{j} + 5\mathbf{k}.
First, express the known point AA as the position vector a=2iβˆ’j+4k=(2βˆ’14)\mathbf{a} = 2\mathbf{i} - \mathbf{j} + 4\mathbf{k} = \begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix}.
Substitute a\mathbf{a} and d\mathbf{d} directly into the standard form to get r=(2βˆ’14)+Ξ»(3βˆ’25)\mathbf{r} = \begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ -2 \\ 5 \end{pmatrix}, which can also be written as r=(2+3Ξ»)i+(βˆ’1βˆ’2Ξ»)j+(4+5Ξ»)k\mathbf{r} = (2 + 3\lambda)\mathbf{i} + (-1 - 2\lambda)\mathbf{j} + (4 + 5\lambda)\mathbf{k} for λ∈R\lambda \in \mathbb{R}.