Horse and rail: the dot product as work
About this app
A horse pulls a wagon at an angle α: the force splits into a part along the rail that does work and a perpendicular part that is lost. W = |F| · |s| · cos α; at 90° the wagon stands still. This gives the dot product F · s.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
The idea
The dot product of two vectors is a · b = |a| · |b| · cos α, where α is the angle between them. The result is a number, a scalar, hence the German name Skalarprodukt. Physically it describes work: if a force F pulls at an angle to the path s, only its part along the path counts, and that part is |F| · cos α. Perpendicular to the path no work is done. That is why the dot product of two nonzero vectors is zero exactly when they are perpendicular.
What you see
A wagon stands on a straight rail and a horse pulls it at the angle α. The force F splits into a part along the rail and a perpendicular part, which together give F again. As the wagon travels the distance s, the work W = |F_h| · |s| grows. With |F| = 1000 N, |s| = 200 m and α = 60°, that is 100,000 Nm. The values panel computes cos α, both parts and W live.
Things to try
Drag the horse from α = 0° to 90°: the part along the rail shrinks like the cosine, and at 90° the wagon stands still although the horse pulls with full force. Beyond 90° it pulls backwards and the work becomes negative. Change |F| and |s| and check that W grows in proportion to both. In explain mode, the AI tutor guides you through the app.
Frequently asked questions
- What is the dot product intuitively?
- It measures how much of one vector points in the direction of the other, multiplied by that vector’s length: a · b = |a| · |b| · cos α. In physics this is the work W = F · s done by a force along a path. The result is a number, not a vector.
- Why is the dot product zero for perpendicular vectors?
- Because cos 90° = 0. A force perpendicular to the path has no component along the path and does no work. Conversely, if the dot product of two nonzero vectors is zero, they are perpendicular.
- How do you compute the angle between two vectors?
- Use cos α = a · b / (|a| · |b|). The dot product is computed component by component, e.g. (1, 2, 3) · (4, 5, 6) = 4 + 10 + 18 = 32. With |a| = √14 and |b| = √77 this gives cos α ≈ 0.9746 and α ≈ 12.9°.
- What does the Horse and Rail app show?
- It shows a horse pulling a wagon at an angle, the split of the force into a part along the rail and a perpendicular part, and the work W = |F| · |s| · cos α. Angle, force and distance can be adjusted, and the ride can be played.
Subject: Linear algebra
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