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The image shows a problem involving a 3D coordinate system and vectors. There are references to a figure (2-25) and the tasks involve calculating the values for different components of vectors between points \( A \), \( B \), and \( C \). Specifically, the problems seem to ask for certain components of vector \( T \) in the coordinates \( A_B \), \( A_C \), \( B_T \), and \( C_T \).

Based on the structure of the diagram, it seems to deal with vector transformations between coordinate systems. The 30° angle shown in the diagram suggests there is a rotation or transformation between axes. The points seem to correspond to a frame of reference on a cubic structure.

Could you clarify if you need help with solving specific parts of the problem (e.g., finding one of the vector components like \( A_B T \), or transforming vectors between these coordinate systems)? This would help me provide the necessary solution steps.

Do you have any further questions or want a more specific explanation on any part? 

Here are five relative questions:
1. How are vectors between two coordinate systems typically transformed in 3D?
2. What role does the 30° angle play in transforming vectors between systems?
3. How can we calculate components like \( A_B T \) in this context?
4. What type of matrix would you use to handle rotations between coordinate systems?
5. What is the significance of each axis \( \hat{X}, \hat{Y}, \hat{Z} \) in this problem?

**Tip:** When solving transformation problems, always pay attention to angles between axes and how they impact the resulting vector components.

求ABT、ACT、BCT、CAT的值，参见图2-25。

Detailed steps for calculating vector components ABT, ACT, BCT, and CAT using transformations between 3D coordinate systems. Learn the role of angles and transformations in finding these components using trigonometric identities and vector mathematics.

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