https://file.aiquickdraw.com/mathbot/file/172735048487011hfg7ps.jpg

Let's analyze the information from the image.

### Given information:
1. **\(\vec{A}\) is parallel to \(\vec{B}\)**: This means \(\vec{A} = k\vec{B}\) for some scalar \(k\), where \(k > 0\) (i.e., they have the same direction).
2. **\(\vec{B}\) is anti-parallel to \(\vec{C}\)**: This means \(\vec{B} = -k\vec{C}\) for some scalar \(k\), meaning \(\vec{B}\) and \(\vec{C}\) have opposite directions.
3. **\(\vec{C}\) is the negative of \(\vec{A}\)**: This gives us \(\vec{C} = -\vec{A}\), which means \(\vec{C}\) points in the opposite direction of \(\vec{A}\).
4. **\(\vec{B}\) and \(\vec{D}\) are equal vectors**: So \(\vec{B} = \vec{D}\), meaning they have the same magnitude and direction.

### The question asks: "Which of the following vectors **DO NOT** have the same direction?"

Let's check each option:
- **Option A: \(\vec{A}\) and \(\vec{B}\)**: Since \(\vec{A}\) is parallel to \(\vec{B}\), they have the same direction.
  
- **Option B: \(\vec{B}\) and \(\vec{D}\)**: Given that \(\vec{B}\) and \(\vec{D}\) are equal vectors, they have the same direction.
  
- **Option C: \(\vec{A}\) and \(\vec{D}\)**: Since \(\vec{A}\) is parallel to \(\vec{B}\), and \(\vec{B} = \(\vec{D}\), \(\vec{A}\) and \(\vec{D}\) also have the same direction.
  
- **Option D: \(\vec{C}\) and \(\vec{A}\)**: Since \(\vec{C} = -\vec{A}\), they are in opposite directions.

### Conclusion:
The correct answer is **Option D** (\(\vec{C}\) and \(\vec{A}\)) because they do not have the same direction.

Would you like further details on the solution? Here are five additional questions to consider:

1. What does it mean for two vectors to be anti-parallel?
2. How can we represent the relationship between parallel vectors algebraically?
3. Can two vectors with different magnitudes be considered parallel?
4. What role does the scalar \(k\) play in defining vector direction?
5. How would this problem change if one of the vectors had a different magnitude?

**Tip:** When vectors are anti-parallel, their directions are exactly opposite, but they may still have the same magnitude.

Which of the following vectors do not have the same direction?

Learn how to identify which vectors do not have the same direction using the concepts of parallel and anti-parallel vectors. This problem covers key vector algebra concepts, including relationships between vectors. Suitable for students in Grades 9-12.

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