you are wrong. The option C says 462pi/6 .not 462pi/462

Thank you for pointing that out! Let's correct the mistake and go through the problem again carefully.

### Given Coordinates:
- Point \( a = (1, 0) \)
- Point \( b = (0, 0) \)
- Point \( c = (-1, 0) \)

### Vectors:
- Vector \( \overrightarrow{ba} = (1, 0) \)
- Vector \( \overrightarrow{bc} = (-1, 0) \)

### Angle Calculation:
We want to find the angle \( \theta = \angle abc \).

1. **Dot Product**:
   \[
   \overrightarrow{ba} \cdot \overrightarrow{bc} = (1, 0) \cdot (-1, 0) = 1 \times -1 + 0 \times 0 = -1
   \]

2. **Magnitude of the Vectors**:
   - \( \|\overrightarrow{ba}\| = \sqrt{1^2 + 0^2} = 1 \)
   - \( \|\overrightarrow{bc}\| = \sqrt{(-1)^2 + 0^2} = 1 \)

3. **Cosine of the Angle**:
   \[
   \cos \theta = \frac{-1}{1 \times 1} = -1
   \]
   \[
   \theta = \cos^{-1}(-1) = \pi \text{ radians}
   \]

### Checking Against Options:
Now, let's check which of the options matches \(\pi\) radians.

- Option A: \( \frac{456\pi}{6} = 76\pi \) (Not equal to \(\pi\))
- Option B: \( \frac{459\pi}{6} = 76.5\pi \) (Not equal to \(\pi\))
- Option C: \( \frac{462\pi}{6} = 77\pi \) (Not equal to \(\pi\))
- Option D: \( \frac{468\pi}{6} = 78\pi \) (Not equal to \(\pi\))

Upon reviewing the given options, it appears none of the choices match exactly \(\pi\) radians. This suggests a possible error in the options provided or misinterpretation. However, from our calculation, the correct angle between these two vectors is definitely \(\pi\) radians.

### Conclusion:
The correct angle measure is \(\pi\) radians, but none of the provided options match this exact value.

Would you like to discuss this further, or have any other questions?

Here are some follow-up questions you might consider:
1. How to deal with incorrect or misleading options in multiple-choice questions?
2. What are the common mistakes when calculating vector angles?
3. How would you convert radians to degrees?
4. What does a \(\pi\) radians angle look like on the unit circle?
5. How to find angles between vectors in 3D?
6. Why is it important to verify the cosine value before using the inverse function?
7. What is the interpretation of the dot product when the angle is \(\pi\) radians?
8. How do options in standardized tests typically relate to correct answers?

**Tip:** Always double-check the values and consider the possibility of misprints or errors in test questions.

Learn how to calculate the angle between vectors (1, 0) and (-1, 0) with a detailed step-by-step solution. This problem covers key geometric concepts including vector operations, dot product calculation, and cosine theorem. Suitable for advanced high school students.

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