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The problem: When translating the components of a vector between two or more coordinate systems, it's incumbent to identify those directional cosines which perform such a task, and furthermore to do so between reciprocal vector bases where each vector in Euclidean 3 - space ( later in Riemann N-space ) expressed in one basis is orthogonal ( i.e., perpendicular ) to two other vectors taken in the other ( reciprocal ) basis.
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