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Chapter 14  Final Thoughts

14.1  Dot and cross products

Multiplying a vector by a scalar is a straightforward operation; so is adding two vectors. But multiplying two vectors is more subtle. It turns out that there are two vector operations that resemble multiplication: the dot product and the cross product.

The dot product of vectors u and v is a scalar:

u · v = | u | | v | cosθ     (1)

where θ is the smallest angle between u and v. We already know how to compute magnitudes, and you could probably figure out how to compute θ, but you don’t have to. MATLAB provides a function, dot, that computes dot products:

d = dot(U, V)

dot works in any number of dimensions, as long as U and V have the same number of elements.

If one of the operands is a unit vector, you can use the dot product to compute the component of a vector u that is in the direction of a unit vector, r:

s = dot(U, Rhat)

In this example, s is the scalar projection of u onto r. The vector projection is the vector that has magnitude s in the direction of r:

V = dot(U, Rhat) * Rhat

The cross product of vectors u and v is a vector whose direction is perpendicular to u and v (with orientation determined by the right-hand rule1) and whose magnitude is

u × v | = | u | | v | sinθ     (2)

where (again) θ is the smallest angle between u and v. MATLAB provides a function, cross, that computes cross products.

C = cross(U, V)

cross calculates the cross product of corresponding vectors along the first array dimension whose size equals 3.

A common use of cross is to compute torques. If you represent a moment arm R and a force F as vectors with size equal to 3, then the torque is just

Tau = cross(R, F)

If the components of R are in meters and the components of F are in Newtons, then the torques in Tau are in Newton-meters.

14.2  What is a model for?

In Section 8.2 I defined a “model” as a simplified description of a physical system, and said that a good model lends itself to analysis and simulation, and makes predictions that are good enough for the intended purpose.

Since then, we have seen a number of examples; now we can say more about what models are for. The goals of a model tend to fall into three categories.

Some models make predictions about physical systems. As a simple example, the duck model in Exercise 2 predicts the level a duck floats at. At the other end of the spectrum, global climate models try to predict the weather tens or hundreds of years in the future.
Models are useful for engineering design, especially for testing the feasibility of a design and for optimization. For example, in Exercise 6 you were asked to design the golf swing with the perfect combination of launch angle, velocity and spin.
Models can answer scientific questions. For example, the Lotka-Volterra model in Section 10.4 offers a possible explanation of the dynamics of animal populations systems in terms of interactions between predator and prey species.

The exercises at the end of this chapter include one model of each type.

14.3  Glossary

dot product:
A scalar product of two vectors, proportional to the norms of the vectors and the cosine of the smallest angle between them.
cross product:
A vector product of two vectors with norm proportional to the norms of the vectors and the sine of the angle between them, and direction perpendicular to both.
The component of one vector that is in the direction of the other (might be used to mean “scalar projection” or “vector projection”).

14.4  Exercises

Exercise 1   If you put two identical bowls of water into a freezer, one at room temperature and one boiling, which one freezes first?

Hint: you might want to do some research on the Mpemba effect.

Exercise 2   You have been asked to design a new skateboard ramp; unlike a typical skateboard ramp, this one is free to pivot about a support point. Skateboarders approach the ramp on a flat surface and then coast up the ramp; they are not allowed to put their feet down while on the ramp. If they go fast enough, the ramp will rotate and they will gracefully ride down the rotating ramp. Technical and artistic display will be assessed by the usual panel of talented judges.

Your job is to design a ramp that will allow a rider to accomplish this feat, and to create a physical model of the system, a simulation that computes the behavior of a rider on the ramp, and an animation of the result.

Exercise 3  

A binary star system contains two stars orbiting each other and sometimes planets that orbit one or both stars.2 In a binary system, some orbits are “stable” in the sense that a planet can stay in orbit without crashing into one of the stars or flying off into space.

Simulation is a useful tool for investigating the nature of these orbits, as in Holman, M.J. and P.A. Wiegert, 1999, “Long-Term Stability of Planets in Binary Systems,” Astronomical Journal 117, available from

Read this paper and then modify your planetary simulation to replicate or extend the results.


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