Oliver Heaviside in England and Josiah Willard Gibbs, a professor at Yale University in Connecticut, also felt that quaternion methods were too cumbersome, often requiring the scalar or vector part of a result to be extracted.
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Gibbs's notation and the name "cross product" later reached a wide audience through Vector Analysis, a textbook by Edwin Bidwell Wilson, a former student.
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It is also used to describe the Lorentz force experienced by a moving electrical charge in a magnetic field.
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Around this time Josiah Willard Gibbs and Oliver Heaviside developed vector calculus, which included separate cross product and dot products that were derived from quaternion multiplication.