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  • Am I right in saying that accelerations can only be given a sign if a coordinate system is defined in relation to which they are described?
  • Is this idea applied to any vector quantity?
  • Is there a guideline for this?
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More or less.

Velocity, accelerations are vector quantities. You can get a scalar quantity related to a direction taking the dot product of the vector quantity $\mathbf{v}$ of interest and a unit-length vector $\mathbf{\hat{t}}$ in the desired direction,

$$v_t = \mathbf{v} \cdot \mathbf{\hat{t}} \ .$$

If you introduce a base, or a set of coordinates inducing a base, you could perform (be careful with non-Cartesian coordinates) projection of the vector over the vectors of this base.

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