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There are various physical situations in which the cross product naturally arises, for example in various relationships between electricity and magnetism. Another example is torque (the rotational equivalent of "force"): torque depends on the distance from the reference point and on the force. It also depends on the angle between the two (including the direction in the "distance"). Finally, the torque can conveniently be defined as having a "direction" that points in the axis of the resulting rotation (or angular acceleration). This gives you all the characteristics of a cross product.

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Q: What is the physical significance of cross product?
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