By W. Lauterborn, T. Kurz
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Additional info for Coherent Optics - Fundamentals and Applications
It is represented by a cross between vectors, C = A × B. 45) where θ is again the angle between the vectors A and B. 5 Illustration of the cross product. Vector A points in the +ˆy direction, while B points in the −ˆx direction. Then C = A × B points in the +ˆz direction, while C = B × A points in the −ˆz direction. A × A = 0 for any vector A. The parallelogram formed by A and B deﬁnes a plane, and we assign to the vector C a direction such that it is perpendicular to the plane of A and B in a manner such that A, B, and C form a right-handed system.
It follows immediately from the deﬁnition of the direction of the cross product that A · C = A · (A × B) = 0. 53) In other words, the cross product of two vectors is orthogonal (perpendicular) to each of the vectors individually. This can also be readily shown by using the component deﬁnitions of the dot and cross products. 4 Useful products of vectors Two types of multiple product of vectors appear quite regularly in physical problems; we brieﬂy discuss each of them. 1 Triple scalar product The triple scalar product is given by A · (B × C).
4 Useful products of vectors Two types of multiple product of vectors appear quite regularly in physical problems; we brieﬂy discuss each of them. 1 Triple scalar product The triple scalar product is given by A · (B × C). 54) It can readily be shown that the triple scalar product may be written in terms of components as A · (B × C) = Ax (By Cz − Bz Cy ) + Ay (Bz Cx − Bx Cz ) + Az (Bx Cy − By Cx ). 55) From this result, the following symmetry properties of the triple scalar product can be deduced: A · (B × C) = B · (C × A) = C · (A × B).