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Vector Dot Product: Single vs. Multiple Coordinate Frames

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Published on May 31, 2013

We revisit the computation of the dot product of two vectors.

In the simple case, with a single coordinate frame, the usual approach of summing the product of corresponding vector components is sufficient. However, this breaks when there are multiple coordinate frames.

With vectors expressed in multiple coordinate frames, it is important to write out the expanded dot product explicitly. Then, the individual dot products between unit vectors can be obtained from the components of the rotation matrix relating the frames.

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