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I am familiar with using cross products in physics to answer questions about force and torque, but are there applications to other scientific fields? The examples that I've seen in most calculus textbooks deal only with physics or engineering questions.

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+1 I am hoping we see some neat ones, however I agree with the answer below that since this is a Euclidean phenomenon, you are probably going to want exterior products, of which I expect appications will be more plentiful/redibly available. – BBischof Feb 21 '11 at 20:54
This is a list-of-applications question, so should be CW. – Willie Wong Feb 22 '11 at 10:23
up vote 4 down vote accepted

To "questions about force and torque" I would add the purely geometrical field of ${\it kinematics}$ which deals with the movement of three-dimensional bodies in three-space, and there are applications in elementary three-dimensional geometry, as determining the distance between two lines in three-space. Then there is a simple rule for solving two homogeneous linear equations in three unknowns: The general solution of the system $$a_1 x_1+a_2 x_2+a_3 x_3=0, \quad b_1 x_1+b_2 x_2+b_3 x_3=0$$ is given by $\ {\bf x}=\lambda\ ({\bf a}\times {\bf b})$, $\ \lambda\in{\mathbb R}$. But otherwise no application of the cross product comes to mind, the reason being that this concept is strictly confined to euclidean spaces of dimension three. Only for $d=3$ a skew bilinear form can be represented by a vector, i.e. an element of the base space.

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