Pulling a Sled: The Part of the Force That Counts

Pull a sled with a rope at an angle and only part of your pull moves it forward; the rest lifts or pushes sideways. That useful part is the projection of the force onto the direction of motion, and its size is the dot product.

How it works

Dot products and orthogonal projection

In one sentence

$\mathbf{u}\cdot\mathbf{v} = \sum u_iv_i = \|\mathbf{u}\|\|\mathbf{v}\|\cos\phi$ ; vectors are orthogonal when $\mathbf{u}\cdot\mathbf{v} = 0$ , and the orthogonal projection of $\mathbf{y}$ onto $\mathbf{u}$ is $\hat{\mathbf{y}} = \frac{\mathbf{y}\cdot\mathbf{u}}{\mathbf{u}\cdot\mathbf{u}}\mathbf{u}$ , with $\mathbf{y} - \hat{\mathbf{y}} \perp \mathbf{u}$ .

General formula

\operatorname{proj}_{{\htmlData{sym=1}{\mathbf{h}}}}{\htmlData{sym=0}{\mathbf{w}}} = \frac{{\htmlData{sym=0}{\mathbf{w}}}\cdot{\htmlData{sym=1}{\mathbf{h}}}}{{\htmlData{sym=1}{\mathbf{h}}}\cdot{\htmlData{sym=1}{\mathbf{h}}}}\,{\htmlData{sym=1}{\mathbf{h}}}, \qquad {\htmlData{sym=0}{\mathbf{w}}}\cdot{\htmlData{sym=1}{\mathbf{h}}} = \|{\htmlData{sym=0}{\mathbf{w}}}\|\|{\htmlData{sym=1}{\mathbf{h}}}\|\cos{\htmlData{sym=2}{\phi}}

Shine a light straight down onto the line of motion: the force's shadow on it is the projection, and the shadow's length is the dot product.

What the symbols mean

$\mathbf{w}$
Rope force
Your pull, along the rope.
$\mathbf{h}$
Direction of motion
Unit vector along the ground.
$\phi$
Angle
Between the rope and the ground.

The rope pulls with force $\mathbf{w}$ ; the sled can only move along the direction $\mathbf{h}$ .
The dot product $\mathbf{w}\cdot\mathbf{h} = \|\mathbf{w}\|\|\mathbf{h}\|\cos\phi$ measures how much of $\mathbf{w}$ points along $\mathbf{h}$ .
The projection $\frac{\mathbf{w}\cdot\mathbf{h}}{\mathbf{h}\cdot\mathbf{h}}\mathbf{h}$ is that part as a vector; what's left, $\mathbf{w} - \operatorname{proj}$ , is perpendicular to the motion.
Work done is (force along the motion) × distance, so a steeper rope wastes more effort lifting.

Learn the math

Inner Product, Length and OrthogonalityLin Alg · Lesson 31, Orthogonality and Least Squares Orthogonal ProjectionsLin Alg · Lesson 33, Orthogonality and Least Squares