Steering a Hovercraft with Two Thrusters

A hovercraft has two fixed thrusters, each pushing along its own direction, forward or reverse. To push the craft any way you like, you need some mix of the two: a linear combination. If the thrusters point along the same line, no mix can push you sideways: they're linearly dependent, and their span is just a line.

How it works

Linear combinations, span and independence

In one sentence

Vectors $\mathbf{v}_1, \ldots, \mathbf{v}_k$ are linearly independent if $c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k = \mathbf{0}$ only when every $c_i = 0$ . Their span is the set of all linear combinations; in $\mathbb{R}^2$ , two vectors span the plane exactly when they're independent.

General formula

{\htmlData{sym=0}{m}}\,{\htmlData{sym=2}{\mathbf{u}}} + {\htmlData{sym=1}{n}}\,{\htmlData{sym=3}{\mathbf{v}}} = {\htmlData{sym=4}{\mathbf{f}}}, \qquad \text{solvable for every } {\htmlData{sym=4}{\mathbf{f}}} \iff {\htmlData{sym=2}{\mathbf{u}}}, {\htmlData{sym=3}{\mathbf{v}}} \text{ independent} \iff \det\begin{bmatrix}{\htmlData{sym=2}{\mathbf{u}}} & {\htmlData{sym=3}{\mathbf{v}}}\end{bmatrix} \ne 0

Independent directions each add something new. Parallel ones repeat each other, so together they still reach only one line.

What the symbols mean

$m$
Thruster 1 strength
How hard thruster 1 pushes (negative: reverse).
$n$
Thruster 2 strength
How hard thruster 2 pushes.
$\mathbf{u}$
Thruster 1 direction
Unit vector along thruster 1.
$\mathbf{v}$
Thruster 2 direction
Unit vector along thruster 2.
$\mathbf{f}$
Desired push
The force you want on the craft.

Thruster 1 pushes along the unit vector $\mathbf{u}$ with strength $m$ ; thruster 2 along $\mathbf{v}$ with strength $n$ (negative means reverse).
The total push is the linear combination $m\mathbf{u} + n\mathbf{v}$ : tip-to-tail arrows.
To get a desired force $\mathbf{f}$ , solve the vector equation for $m$ and $n$ : two equations, two unknowns.
If $\mathbf{u}$ and $\mathbf{v}$ are independent their span is the whole plane, so any $\mathbf{f}$ works. If they're parallel, only forces along that line can be made.

Learn the math

Vector EquationsLin Alg · Lesson 3, Linear Equations in Linear Algebra Linear IndependenceLin Alg · Lesson 6, Linear Equations in Linear Algebra Subspaces of RⁿLin Alg · Lesson 13, Matrix Algebra

Steering a Hovercraft with Two Thrusters