What are practical examples of a direction cosine vector/matrix?

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I think I understand what a direction cosine vector of a vector. But I'm struggling to:

Understand how these are used practically when it comes to rotation etc? How it this better or worse that a rotation matrix?

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If you have a series of rotation matrices that get you where you want to go, the matrix with all of the direction cosines in it will get you there in one shot. It's just the matrix product of the individual rotations.